The period-integral description of tautological-system solutions for complete intersections

Let $X=G/P$ be a homogeneous variety, let $E=L_1\oplus\cdots\oplus L_r$ be a direct sum of homogeneous line bundles, and let $Y_i$ be the zero locus of the $L_i$-component of a section $s_b$ corresponding to $b\in V^\vee$. Write $\tau_V$ for the tautological system and $\mathcal D_{V^\vee}$ for the sheaf of differential operators on $V^\vee$. The solution sheaf is evaluated at $b$, and $H_n(X-(Y_1\cup\cdots\cup Y_r))$ denotes the corresponding homology group. The period-integral conjecture. There is a natural isomorphism

$$\operatorname{Hom}_{\mathcal D_{V^\vee}}(\tau_V,\mathcal O)|_b\cong H_n(X-(Y_1\cup\cdots\cup Y_r)).$$

This conjecture proposes that solutions of the tautological system for the complete intersection are represented by period integrals over cycles in the complement of the divisors $Y_1,\ldots,Y_r$. The source gives no resolution status, so the conjecture is treated as open.