I have a conjecture that there exist five parameter values (presumably, integers or half-integers), $i_1,i_2,i_3,i_4,i_5$ for which the function (the hypergeometric function being "regularized", following Mathematica's treatment), \begin{equation} \frac{ \Gamma (d+1)^3 \, _3\tilde{F}_2\left(k i_1-\frac{d}{2},\frac{d}{2}+k i_2,d+k i_3;\frac{d}{2}+k i_4+1,\frac{3 d}{2}+k i_5+1;e^2\right)}{2 \Gamma \left(\frac{d}{2}+1\right)^2}, \end{equation} yields for $d=2, k=1$, \begin{equation} \frac{1}{10} e^2 \left(2 \left(e^2-6\right) e^2+15\right), \end{equation} for $d=4,k=2$, \begin{equation} \frac{1}{462} \left(175 e^8-1374 e^6+4290 e^4-5830 e^2+2970\right), \end{equation} for $d=6,k=3$, \begin{equation} \frac{17930 e^{12}-188865 e^{10}+854658 e^8-2113032 e^6+2952288 e^4-2197692 e^2+686868}{24310}, \end{equation} and for, further say (in the interest of manageably-sized expressions), $d=2,k=3$, \begin{equation} \frac{1}{252} \left(41 e^8-410 e^6+1215 e^4-1560 e^2+840\right) \end{equation} and, say, for $d=4,k=4$, \begin{equation} \frac{4851 e^{12}-54976 e^{10}+265160 e^8-656320 e^6+888160 e^4-640640 e^2+200200}{12870}. \end{equation} The integration formula I employed to yield these five polynomials was presented in the recent posting
(The results above are given by those, but now divided by $e^d \equiv \varepsilon^d$, yielded by the integration formula given there. Of course, $e \in [0,1]$ is a variable, and not the natural log base--the notation serving as a convenient replacement for an original $\varepsilon$.)
If the conjecture is true, presumably the five parameters $i_1,i_2,i_3,i_4,i_5$ can be determined by the five polynomials above, but more can be readily generated (and used for further testing).
For odd values of $d$, polynomial expressions are not obtained.
The question stated here pertains to the issue discussed in sec. IX.B of my posting, https://arxiv.org/abs/1803.10680, "Qubit-qudit separability/PPT-probability investigations, including Lovas-Andai formula advancements", of finding an “extended Lovas-Andai master formula”, denoted there by $ \tilde{\chi}_{d,k}(\varepsilon)$.
It seems now that the conjecture (almost assuredly, technically incorrect) is not the natural way to approach the underlying/basic problem. Charles F. Dunkl has provided me with an analysis in which the solution is given in the form of a summation over the index $j$ from 0 to $k$, of terms involving 3F2 hypergeometric functions. The text of his answer has been posted as an answer to https://mathematica.stackexchange.com/questions/171351/evaluate-over-a-two-dimensional-domain-the-integral-of-hypergeometric-based-f/173532#173532