Canonical-bundle conjecture for the divisibility-one fixed locus

Let $(X,\lambda)$ be a polarized irreducible holomorphic symplectic manifold of $\mathrm{K3}^{[n]}$-type with $q_X(\lambda)=2$ and $\operatorname{div}(\lambda)=1$. Let $\tau_{\lambda}$ be the associated involution, and set

$$F:=\operatorname{Fix}(\tau_{\lambda}).$$

Let $L$ be the ample line bundle such that $c_1(L)=\lambda$. Canonical-bundle conjecture. In $\operatorname{Pic}(F)$,

$$\omega_F^{\otimes n}=L|_F^{\otimes \frac{(n+2)(n+1)}{2}}.$$

This is the conjectural canonical-bundle formula for the irreducible fixed locus in the divisibility-one case; the source gives no evidence that it has been proved or disproved.