Purity and exterior-power basis conjecture for the hypergeometric cohomology of $Y$

Let $w\in Z_{r+1}$, and let $(\Omega^{\bullet}(Y),\nabla_{\tilde{\omega}})$ be the complex whose degree-$r$ cohomology contains the classes represented by the forms

$$\psi_{j_{1}}\wedge\cdots\wedge\psi_{j_{r}},\quad 1\leq j_{1}<\cdots<j_{r}\leq N-2.$$

Here the $1$-forms $\psi_{1},\dots,\psi_{N-2}\in\Omega^{1}(Y)$ depend holomorphically on $w$.

Purity and exterior-power basis conjecture. For every $w\in Z_{r+1}$, the cohomology of $(\Omega^{\bullet}(Y),\nabla_{\tilde{\omega}})$ is pure,

$$\dim_{\mathbb{C}}H^{r}(\Omega^{\bullet}(Y),\nabla_{\tilde{\omega}})=\binom{N-2}{r},$$

and the displayed $r$-forms give a basis of $H^{r}(\Omega^{\bullet}(Y),\nabla_{\tilde{\omega}})$.

The preceding discussion establishes the basis assertion for Veronese points and linear independence on a Zariski-open subset containing the Veronese image. The conjecture extends this conclusion to every point of $Z_{r+1}$ and asserts purity and the stated dimension throughout.