Let $X$ be a hyperkähler variety.

In an [article](https://arxiv.org/abs/math/0305042) (Conjecture 2.1) from some years ago, Markman conjectured that any monodromy operator acting trivially on $H^2(X,\mathbb Z)$ is the identity operator, at least if $X$ is deformation equivalent to the Hilbert scheme of length $n$ subschemes of a K3 surface.

This conjecture was proven true by Markman himself for certain values of $n$ (and "almost" true for all other values) in [this](https://arxiv.org/abs/math/0601304) subsequent paper (Corollary 1.6).

> (1) What is the current status of this conjecture?

> (2) Is it known to be false for certain deformation equivalence classes of hyperkähler varieties?