Definitions and notation.
Fix $1<p<\infty$, and define the (reflexive) Banach space \begin{equation*}E=\left(\bigoplus_{n=1}^\infty\ell_\infty^{2^{n-1}}\right)_{\ell_p}.\end{equation*} It has a "canonical" unit vector basis $((e_k^{(2^{n-1})})_{k=1}^{2^{n-1}})_{n=1}^\infty$. Let $(e_j)_{j=1}^\infty$ denote this same basis in the same order, when we want to use a single index $j$ instead of two indices $(k,n)$.
Suppose that $X$ is another Banach space with an unconditional normalized basis $((x_k^{(2^{n-1})})_{k=1}^{2^{n-1}})_{n=1}^\infty=(x_j)_{j=1}^\infty$, and that there is a constant $C\in[1,\infty)$ such that the following two properties are satisfied.
\begin{equation}\tag{1}\|\sum_{j=1}^Nx_j\|_X\leq C\|\sum_{j=1}^Ne_j\|_E\;\;\;\text{ for all }1\leq N<\infty,\end{equation} and \begin{equation}\tag{2}\frac{1}{C}\|\sum_{j=1}^\infty a_je_j\|_E\leq \|\sum_{j=1}^\infty a_jx_j\|_X\;\;\;\text{ for all }(a_j)_{j=1}^\infty\in c_{00},\end{equation} where $c_{00}$ denotes the space of scalar sequences with finitely many nonzero entries.
Question. Is there a constant $C'\in[1,\infty)$ such that \begin{equation}\tag{3}\|\sum_{j=1}^\infty a_jx_j\|_X\leq C'\|\sum_{j=1}^\infty a_je_j\|_E\;\;\;\text{ for all }(a_j)_{j=1}^\infty\in c_{00}?\end{equation} In other words, is $(x_j)_{j=1}^\infty$ equivalent to $(e_j)_{j=1}^\infty$?
Discussion.
It is clear (see, for instance, this question) that each $(x_k^{(2^{n-1})})_{k=1}^{2^{n-1}}$ is uniformly equivalent to $(e_k^{(2^{n-1})})_{k=1}^{2^{n-1}}$ (the standard basis for $\ell_\infty^{2^{n-1}}$). Furthermore, in the special case I am looking at, I know that $X$ is actually a quotient of $E$. So, there are plenty of clues to suggest the answer is "yes." Nevertheless, I have not yet been able to prove it one way or the other.
Thanks!