Lunts's equality conjecture for hyperkähler manifolds

Let $X$ be a projective hyperkähler manifold. The Lunts equality conjecture. There is an equality of Lie subalgebras of $\operatorname{End}(H^\bullet(X,\mathbb{Q}))$:

$$L^{eq}(X)=\mathfrak{g}_{NS}(X).$$

This conjecture proposes that the Lie algebra generated by derived autoequivalences agrees with the Neron-Severi Lie algebra for projective hyperkähler manifolds. The paper discusses evidence including one inclusion in a weak version, but does not establish the full equality in general.