Theta-vanishing conjecture for isomonodromy actions

Let $f:X\to S$ be a smooth projective morphism of smooth complex varieties, and fix $s\in S$. Let $\Theta_{X/S}$ be the quantity on the Dolbeault moduli space appearing in the source. Theta-vanishing conjecture. If $\Theta_{X/S}\equiv 0$ on $\mathscr{M}_{\operatorname{Dol}}(X/S,r)$, then the action of $\pi_1(S(\mathbb{C})^{\an},s)$ on $M_B(X_s,r)(\mathbb{C})$ factors through a finite group. The conjecture is a complex-algebraic question; the source notes that for relative curves the corresponding vanishing implies that $X/S$ is isotrivial, while the general statement is reduced to an arithmetic integral-points question and remains open.