Beckmann's hard Lefschetz conjecture for stable atomic sheaves

Let $X$ be the hyper-Kähler variety under consideration, let $E$ be a stable atomic sheaf on $X$, and let $\operatorname{obs}_E(\eta)\in \operatorname{Ext}^2(E,E)$ be a nonzero element in the image of the obstruction map.

Beckmann's conjecture. The element $\operatorname{obs}_E(\eta)$ has the hard Lefschetz property for the Yoneda product on $\operatorname{Ext}^{\bullet}(E,E)$.

The conjecture is motivated by the aim of constructing symplectic moduli spaces of $1$-obstructed objects. The source does not state whether it has been resolved.