For $n\in\mathbb{N}\land n\ge2$, define the so-called Ising-class integral of the third kind, $E_{n}$, via the sequence of $\left(n-1\right)$-dimensional integrals $$E_{n}:=2\int_{\left[0,1\right]^{n-1}}\prod_{\ell=2}^{n}\mathrm{d}t_{\ell}\,\left(\prod_{1\le j<k\le n}\frac{u_{k}-u_{j}}{u_{k}+u_{j}}\right)^{2},\tag{1}$$ where $u_{k}:=\prod_{i=1}^{k}t_{i}$.
In a paper by David Bailey (see article 96 on this list), it is stated that while closed form expressions are known only for the $2\le n\le4$ cases, a conjectured value for the $n=5$ case has been obtained experimentally:
$$\begin{align} E_{5} &\stackrel{?}{=}42-40\ln{\left(2\right)}+464\ln^{2}{\left(2\right)}-124\,\zeta{\left(2\right)}\\ &~~~~~+80\ln{\left(2\right)}\,\zeta{\left(2\right)}-74\,\zeta{\left(3\right)}\\ &~~~~~+88\ln^{4}{\left(2\right)}+240\ln^{2}{\left(2\right)}\,\zeta{\left(2\right)}-1272\ln{\left(2\right)}\,\zeta{\left(3\right)}\\ &~~~~~+1701\,\zeta{\left(4\right)}-1984\operatorname{Li}_{4}{\left(\frac12\right)}\\ &\approx0.003493653711729521740688067279184251569632944955.\tag{2}\\ \end{align}$$
The conjectured value for $E_{5}$ given above has been found to agree numerically with the value of the integral $(1)$ out to at least $240$ digits.
Question: Can a closed form expression for $E_{5}$ be found? It would suffice to prove that $(2)$ is correct.
Note: To avoid cumbersome subscripts, rename the variables as $t_{2}=:t\land t_{3}=:u\land t_{4}=:v\land t_{5}=:w$. Then, we obtain the alternative expression of $E_{5}$ as the iterated integral
$$\begin{align} E_{5} &=2\int_{0}^{1}\mathrm{d}t\int_{0}^{1}\mathrm{d}u\int_{0}^{1}\mathrm{d}v\int_{0}^{1}\mathrm{d}w\,\left(\frac{1-t}{1+t}\right)^{2}\left(\frac{1-tu}{1+tu}\right)^{2}\\ &~~~~~\times\left(\frac{1-u}{1+u}\right)^{2}\left(\frac{1-tuv}{1+tuv}\right)^{2}\left(\frac{1-uv}{1+uv}\right)^{2}\left(\frac{1-v}{1+v}\right)^{2}\\ &~~~~~\times\left(\frac{1-tuvw}{1+tuvw}\right)^{2}\left(\frac{1-uvw}{1+uvw}\right)^{2}\left(\frac{1-vw}{1+vw}\right)^{2}\left(\frac{1-w}{1+w}\right)^{2}.\tag{3}\\ \end{align}$$
Proof:
$$\begin{align} E_{5} &=2\int_{\left[0,1\right]^{4}}\prod_{\ell=2}^{5}\mathrm{d}t_{\ell}\,\left(\prod_{1\le j<k\le5}\frac{u_{k}-u_{j}}{u_{k}+u_{j}}\right)^{2}\\ &=2\int_{\left[0,1\right]^{4}}\mathrm{d}t_{2}\,\mathrm{d}t_{3}\,\mathrm{d}t_{4}\,\mathrm{d}t_{5}\,\left(\prod_{1\le j<k\le5}\frac{u_{k}-u_{j}}{u_{k}+u_{j}}\right)^{2}\\ &=2\int_{\left[0,1\right]^{4}}\mathrm{d}t_{2}\,\mathrm{d}t_{3}\,\mathrm{d}t_{4}\,\mathrm{d}t_{5}\,\left(\prod_{1\le j<k\le4}\frac{u_{k}-u_{j}}{u_{k}+u_{j}}\right)^{2}\left(\prod_{j=1}^{4}\frac{u_{5}-u_{j}}{u_{5}+u_{j}}\right)^{2}\\ &=2\int_{\left[0,1\right]^{4}}\mathrm{d}t_{2}\,\mathrm{d}t_{3}\,\mathrm{d}t_{4}\,\mathrm{d}t_{5}\,\left(\prod_{1\le j<k\le3}\frac{u_{k}-u_{j}}{u_{k}+u_{j}}\right)^{2}\left(\prod_{j=1}^{3}\frac{u_{4}-u_{j}}{u_{4}+u_{j}}\right)^{2}\\ &~~~~~\times\left(\prod_{j=1}^{4}\frac{u_{5}-u_{j}}{u_{5}+u_{j}}\right)^{2}\\ &=2\int_{\left[0,1\right]^{4}}\mathrm{d}t_{2}\,\mathrm{d}t_{3}\,\mathrm{d}t_{4}\,\mathrm{d}t_{5}\,\left(\prod_{1\le j<k\le2}\frac{u_{k}-u_{j}}{u_{k}+u_{j}}\right)^{2}\left(\prod_{j=1}^{2}\frac{u_{3}-u_{j}}{u_{3}+u_{j}}\right)^{2}\\ &~~~~~\times\left(\prod_{j=1}^{3}\frac{u_{4}-u_{j}}{u_{4}+u_{j}}\right)^{2}\left(\prod_{j=1}^{4}\frac{u_{5}-u_{j}}{u_{5}+u_{j}}\right)^{2}\\ &=2\int_{\left[0,1\right]^{4}}\mathrm{d}t_{2}\,\mathrm{d}t_{3}\,\mathrm{d}t_{4}\,\mathrm{d}t_{5}\,\left(\frac{u_{2}-u_{1}}{u_{2}+u_{1}}\right)^{2}\left(\frac{u_{3}-u_{1}}{u_{3}+u_{1}}\right)^{2}\left(\frac{u_{3}-u_{2}}{u_{3}+u_{2}}\right)^{2}\\ &~~~~~\times\left(\frac{u_{4}-u_{1}}{u_{4}+u_{1}}\right)^{2}\left(\frac{u_{4}-u_{2}}{u_{4}+u_{2}}\right)^{2}\left(\frac{u_{4}-u_{3}}{u_{4}+u_{3}}\right)^{2}\\ &~~~~~\times\left(\frac{u_{5}-u_{1}}{u_{5}+u_{1}}\right)^{2}\left(\frac{u_{5}-u_{2}}{u_{5}+u_{2}}\right)^{2}\left(\frac{u_{5}-u_{3}}{u_{5}+u_{3}}\right)^{2}\left(\frac{u_{5}-u_{4}}{u_{5}+u_{4}}\right)^{2}\\ &=2\int_{\left[0,1\right]^{4}}\mathrm{d}t_{2}\,\mathrm{d}t_{3}\,\mathrm{d}t_{4}\,\mathrm{d}t_{5}\,\left(\frac{t_{1}t_{2}-t_{1}}{t_{1}t_{2}+t_{1}}\right)^{2}\left(\frac{t_{1}t_{2}t_{3}-t_{1}}{t_{1}t_{2}t_{3}+t_{1}}\right)^{2}\left(\frac{t_{1}t_{2}t_{3}-t_{1}t_{2}}{t_{1}t_{2}t_{3}+t_{1}t_{2}}\right)^{2}\\ &~~~~~\times\left(\frac{t_{1}t_{2}t_{3}t_{4}-t_{1}}{t_{1}t_{2}t_{3}t_{4}+t_{1}}\right)^{2}\left(\frac{t_{1}t_{2}t_{3}t_{4}-t_{1}t_{2}}{t_{1}t_{2}t_{3}t_{4}+t_{1}t_{2}}\right)^{2}\left(\frac{t_{1}t_{2}t_{3}t_{4}-t_{1}t_{2}t_{3}}{t_{1}t_{2}t_{3}t_{4}+t_{1}t_{2}t_{3}}\right)^{2}\\ &~~~~~\times\left(\frac{t_{1}t_{2}t_{3}t_{4}t_{5}-t_{1}}{t_{1}t_{2}t_{3}t_{4}t_{5}+t_{1}}\right)^{2}\left(\frac{t_{1}t_{2}t_{3}t_{4}t_{5}-t_{1}t_{2}}{t_{1}t_{2}t_{3}t_{4}t_{5}+t_{1}t_{2}}\right)^{2}\left(\frac{t_{1}t_{2}t_{3}t_{4}t_{5}-t_{1}t_{2}t_{3}}{t_{1}t_{2}t_{3}t_{4}t_{5}+t_{1}t_{2}t_{3}}\right)^{2}\\ &~~~~~\times\left(\frac{t_{1}t_{2}t_{3}t_{4}t_{5}-t_{1}t_{2}t_{3}t_{4}}{t_{1}t_{2}t_{3}t_{4}t_{5}+t_{1}t_{2}t_{3}t_{4}}\right)^{2}\\ &=2\int_{\left[0,1\right]^{4}}\mathrm{d}t_{2}\,\mathrm{d}t_{3}\,\mathrm{d}t_{4}\,\mathrm{d}t_{5}\,\left(\frac{t_{2}-1}{t_{2}+1}\right)^{2}\left(\frac{t_{2}t_{3}-1}{t_{2}t_{3}+1}\right)^{2}\left(\frac{t_{3}-1}{t_{3}+1}\right)^{2}\\ &~~~~~\times\left(\frac{t_{2}t_{3}t_{4}-1}{t_{2}t_{3}t_{4}+1}\right)^{2}\left(\frac{t_{3}t_{4}-1}{t_{3}t_{4}+1}\right)^{2}\left(\frac{t_{4}-1}{t_{4}+1}\right)^{2}\\ &~~~~~\times\left(\frac{t_{2}t_{3}t_{4}t_{5}-1}{t_{2}t_{3}t_{4}t_{5}+1}\right)^{2}\left(\frac{t_{3}t_{4}t_{5}-1}{t_{3}t_{4}t_{5}+1}\right)^{2}\left(\frac{t_{4}t_{5}-1}{t_{4}t_{5}+1}\right)^{2}\left(\frac{t_{5}-1}{t_{5}+1}\right)^{2}\\ &=2\int_{\left[0,1\right]^{4}}\mathrm{d}t\,\mathrm{d}u\,\mathrm{d}v\,\mathrm{d}w\,\left(\frac{t-1}{t+1}\right)^{2}\left(\frac{tu-1}{tu+1}\right)^{2}\left(\frac{u-1}{u+1}\right)^{2}\\ &~~~~~\times\left(\frac{tuv-1}{tuv+1}\right)^{2}\left(\frac{uv-1}{uv+1}\right)^{2}\left(\frac{v-1}{v+1}\right)^{2}\\ &~~~~~\times\left(\frac{tuvw-1}{tuvw+1}\right)^{2}\left(\frac{uvw-1}{uvw+1}\right)^{2}\left(\frac{vw-1}{vw+1}\right)^{2}\left(\frac{w-1}{w+1}\right)^{2}\\ &=2\int_{\left[0,1\right]^{4}}\mathrm{d}t\,\mathrm{d}u\,\mathrm{d}v\,\mathrm{d}w\,\left(\frac{1-t}{1+t}\right)^{2}\left(\frac{1-tu}{1+tu}\right)^{2}\left(\frac{1-u}{1+u}\right)^{2}\\ &~~~~~\times\left(\frac{1-tuv}{1+tuv}\right)^{2}\left(\frac{1-uv}{1+uv}\right)^{2}\left(\frac{1-v}{1+v}\right)^{2}\\ &~~~~~\times\left(\frac{1-tuvw}{1+tuvw}\right)^{2}\left(\frac{1-uvw}{1+uvw}\right)^{2}\left(\frac{1-vw}{1+vw}\right)^{2}\left(\frac{1-w}{1+w}\right)^{2}\\ &=2\int_{0}^{1}\mathrm{d}t\int_{0}^{1}\mathrm{d}u\int_{0}^{1}\mathrm{d}v\int_{0}^{1}\mathrm{d}w\,\left(\frac{1-t}{1+t}\right)^{2}\left(\frac{1-tu}{1+tu}\right)^{2}\\ &~~~~~\times\left(\frac{1-u}{1+u}\right)^{2}\left(\frac{1-tuv}{1+tuv}\right)^{2}\left(\frac{1-uv}{1+uv}\right)^{2}\left(\frac{1-v}{1+v}\right)^{2}\\ &~~~~~\times\left(\frac{1-tuvw}{1+tuvw}\right)^{2}\left(\frac{1-uvw}{1+uvw}\right)^{2}\left(\frac{1-vw}{1+vw}\right)^{2}\left(\frac{1-w}{1+w}\right)^{2}.\blacksquare\\ \end{align}$$