Injectivity of the cohomological monodromy restriction for Hilbert-scheme-type manifolds

Let $X$ be an irreducible holomorphic symplectic manifold deformation equivalent to the Hilbert scheme $S^{[n]}$ of length $n$ subschemes of a K3 surface $S$. Let ${\rm Mon}(X)$ be the monodromy group acting on the integral cohomology of $X$, and let ${\rm Mon}^2(X)$ be its induced action on $H^2(X,\mathbb Z)$.

Monodromy restriction conjecture. The restriction homomorphism

$${\rm Mon}(X)\rightarrow {\rm Mon}^2(X)$$

is injective.

The question appears in the paper's discussion of Torelli questions and local monodromy operators. The supplied text gives no resolution or further evidence, so the conjecture remains open here.