Barnes-function conjecture for the hypergeometric-kernel asymptotic constant

Assume the hypotheses of Proposition~. Let $C$ be the constant in the asymptotic expansions of the hypergeometric-kernel determinant $D(t)$ as $t\to1$, and define

$$G\left[\begin{array}{c}a_1,\ldots,a_m\\b_1,\ldots,b_n\end{array}\right]=\frac{\prod_{k=1}^mG(a_k)}{\prod_{k=1}^nG(b_k)},$$

where $G$ is the Barnes function. Conjectural evaluation of $C$.

$$C=G\left[\begin{array}{c}1-z,1+z,1-z',1+z',1+w,1+w',1+z+z'+w,1+z+z'+w'\\1-z-z',1+z+z',1+z+w,1+z+w',1+z'+w,1+z'+w'\end{array}\right].$$

This extends the previously conjectured zero-magnetic-field formula and is presented as a conjectural evaluation; the paper gives no resolution.