The double EPW sextic realization conjecture for Enriques moduli K3 surfaces

Let $S$ be a polarized K3 surface in $\mathfrak{M}_{\operatorname{BE}}(2t)$ with $t\geq 2$. For some $a$ satisfying $a^2\equiv -1\pmod{t}$, let $\operatorname{M}_{\sigma_{t,a}}(\bar{v}_1)$ and $\operatorname{M}_{\sigma_{t,a}}(\bar{v}_2)$ be the moduli spaces of semistable objects. The double EPW sextic realization conjecture. The spaces $\operatorname{M}_{\sigma_{t,a}}(\bar{v}_1)$ and $\operatorname{M}_{\sigma_{t,a}}(\bar{v}_2)$ can be realized as double EPW sextics in the sense of O'Grady. This is verified for general K3 surfaces, and partly checked for general K3 surfaces in $\mathfrak{M}_{\operatorname{BE}}(2t)_a$ when $a^2-t=-1$ and $t\geq 10$; the general assertion remains open.