Transversality conjecture for the infinitesimal variation of Hodge structures

Let $V$ be a parameter space of smooth projective varieties, let $Y/V$ be the corresponding family, and let the Kodaira-Spencer map be the map referred to as. Let $D_{\check s_{\rm \max},t}$ be the determinantal variety of homomorphisms of rank at most $\check s_{\rm \max}$ in the fiber over $t$, where $\check s_{\rm \max}$ is the integer defined in the paper. Transversality conjecture. If the Kodaira-Spencer map is surjective, then for generic $t\in V$, the map given by the infinitesimal variation of Hodge structures is transversal to $D_{\check s_{\rm \max},t}$, and hence does not intersect it. If true, this would imply $s_{\rm \max}=\check s_{\rm \max}$; the source gives no resolution of the conjecture.