The multiple genus two Cauchy sum leading-term formula

For a positive integer $m$, let $\widetilde X_{12}^{(a)},\widetilde X_{13}^{(a)},\widetilde X_{23}^{(a)}$ for $1\leq a\leq m$ be the formal variables associated with the multiple genus two Cauchy sum, and let $\widetilde\kappa$ be the parameter at which the leading term is taken. Write $\vec i^{(a)}=(i_{12}^{(a)},i_{13}^{(a)},i_{23}^{(a)})$ with all indices nonnegative. The multiple genus two Cauchy sum leading-term formula. The leading term at $\widetilde\lambda=\widetilde\kappa$ is

$$\begin{aligned} \Omega_-&\big(\widetilde X_{12}^{(1)},\widetilde X_{13}^{(1)},\widetilde X_{23}^{(1)},\ldots,\widetilde X_{12}^{(m)},\widetilde X_{13}^{(m)},\widetilde X_{23}^{(m)}\mid\widetilde\kappa\big) \\ &=\sum_{\vec i^{(1)},\ldots,\vec i^{(m)}=0}^{\infty}\prod_{a=1}^{m}\widetilde X_{12}^{i_{12}^{(a)}}\widetilde X_{13}^{i_{13}^{(a)}}\widetilde X_{23}^{i_{23}^{(a)}}(-1)^{i_{23}^{(1)}+\cdots+i_{23}^{(m)}}4^{-\sum_{a=1}^{m}(i_{12}^{(a)}+i_{13}^{(a)}+i_{23}^{(a)})} \\ &\quad\times\frac{\Gamma\left(2\sum_{a=1}^{m}(i_{12}^{(a)}+i_{13}^{(a)}+i_{23}^{(a)})+2\right)\Gamma\left(\frac32\right)^m}{\Gamma\left(\sum_{a=1}^{m}(i_{12}^{(a)}+i_{13}^{(a)}+2i_{23}^{(a)})+2\right)\prod_{a=1}^{m}\Gamma\left(i_{12}^{(a)}+i_{13}^{(a)}+i_{23}^{(a)}+\frac32\right)} \\ &\quad\times\frac{\Gamma\left(\sum_{a=1}^{m}(i_{12}^{(a)}+i_{23}^{(a)})+1\right)\Gamma\left(\sum_{a=1}^{m}(i_{13}^{(a)}+i_{23}^{(a)})+1\right)}{\prod_{a=1}^{m}\Gamma(i_{12}^{(a)}+1)\Gamma(i_{13}^{(a)}+1)\Gamma(i_{23}^{(a)}+1)}. \end{aligned}$$

This series is an $A$-hypergeometric function in the sense of Gelfand, Kapranov, and Zelevinsky. The parser supplies no evidence that the asserted formula has been proved or disproved, so its status remains open.