Holonomic rank conjecture for tautological systems on homogeneous spaces

Let $X$ be an $n$-dimensional projective homogeneous space of a semisimple group $G$, let $V$ be the representation space underlying the tautological system, and let $a\in V^\vee$ define the hypersurface $Y_a\subset X$. Write $\tau=\tau(G,X,\omega_X^{-1},\beta_0)$ for the corresponding tautological system.

Holonomic rank conjecture. The solution rank of $\tau$ at the point $a\in V^\vee$ is given by

$$\dim H_n(X-Y_a).$$

This conjecture addresses when the solution sheaf of the tautological system coincides with the period sheaf and, more generally, compares their ranks. In the supplied source it is proved in the Calabi–Yau case, while the statement is posed for projective homogeneous spaces of semisimple groups in general.