Beckmann's conjecture on simple atomic objects

Let $X$ be the hyper-Kähler variety under consideration, and let $E\in \mathrm{D^b}(X)$ be a simple atomic object. An object is $1$-obstructed when its obstruction map has rank $1$.

Beckmann's conjecture. The object $E$ is $1$-obstructed.

Atomicity is characterized by the cohomological obstruction map having rank $1$, while the source notes that the forward implication to cohomological $1$-obstructedness is known. Beckmann conjectures the converse for simple atomic objects; the source gives no resolution evidence.