Conjecture on the monodromy Hermitian form for A-hypergeometric functions

Let ${\cal I}$ index the vectors and transition-matrix columns $X_I$ arising from the Mellin–Barnes bases, let $H$ be a Hermitian matrix preserved by all monodromy matrices, and let $\Delta_I$ and $\gamma_i^I$ denote the quantities associated with $I\in{\cal I}$ in the construction. The orthogonality relations say that $\overline{X}_J^t H X_I=0$ whenever the corresponding vectors $\v b_I$ and $\v b_J$ have a point in common. Monodromy Hermitian-form conjecture. These orthogonality properties suffice to determine $H$ up to a scalar factor. Furthermore, after a suitable normalization of $H$, for every $I\in{\cal I}$ one has

$$\overline{X}_I H X_I=\Delta_I\prod_{i\notin I}\sin(\pi\gamma_i^I).$$

The conjecture would determine the monodromy-invariant Hermitian form completely from the orthogonality relations and the proposed diagonal values; the source states that these questions had not been answered there.