Barnes-function conjecture for the confluent hypergeometric-kernel constant

Assume that $0\leq\operatorname{Re}(z+z')<1$ and $z,z',w,z+z'+w\notin\mathbb{Z}$, as in Proposition~. Let $C_L$ be the constant in the small-$s$ asymptotics of the confluent hypergeometric-kernel determinant $D_L(s)$. Conjectural evaluation of $C_L$.

$$C_L=\lim_{w'\to\infty}(w')^{-zz'}C=G\left[\begin{array}{c}1-z,1+z,1-z',1+z',1+w,1+z+z'+w\\1-z-z',1+z+z',1+z+w,1+z'+w\end{array}\right].$$

This formula is suggested by the formal large-$w'$ limit of the preceding conjectural constant and is not resolved in the paper.