Extremality conjecture for the variable quantum local system of Fano varieties

Let $X$ be a Fano variety of dimension $n$, and let $\mathbb{V}=\mathbb{V}_f\oplus\mathbb{V}_v$ be the local system associated with its regularised quantum $\mathcal{D}$-module, with variable part $\mathbb{V}_v$. Let $H^{(n/2,n/2)}_{\operatorname{prim}}(X)$ be the space of middle-dimensional primitive Hodge classes on $X$.

Extremality conjecture.

$$\operatorname{rk}\bigl(\operatorname{IH}^1(\mathbb{P}^1,\mathbb{V}_v)\bigr)\leq \dim\bigl(H^{(n/2,n/2)}_{\operatorname{prim}}(X)\bigr).$$

In particular, $\mathbb{V}_v$ is extremal for odd $n$.

Here extremality means that the intersection cohomology group vanishes, equivalently $2\operatorname{rk}(\mathbb{V}_v)=\operatorname{rf}(\mathbb{V}_v)$. The supplied text gives no resolution status.