Let $X$ be a hyperkähler variety.
In an article (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 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?