Boundary-measure conjecture for Fox's H-function

Let $W(x)=\dfrac{\prod_{i=1}^{p}\Gamma(A_i x+a_i)}{\prod_{j=1}^{q}\Gamma(B_j x+b_j)}$, let $\rho$ be the endpoint appearing in its Fox $H$-function representation, and let $\mu=0$. Write $\delta_{\log(1/\rho)}$ for the unit mass at $\log(1/\rho)$. Boundary-measure conjecture. The representing measure is

$$W(x)=\int_{[\log(1/\rho),\infty)}e^{-tx}\left\{A\delta_{\log(1/\rho)}+H_{q,p}^{p,0}\left(e^{-t}\left|\begin{array}{l}(\mathbf B,\mathbf b)\\(\mathbf A,\mathbf a)\end{array}\right.\right)\!\right\}dt,$$

where $A$ is a positive constant.

For $\mu>0$, the paper proves an analogous representation without the boundary atom and proves nonnegativity of the $H$-function under the relevant hypotheses. The conjecture concerns the additional atom required in the limiting case $\mu=0$, by analogy with the corresponding Meijer's $G$-function formula.