Birational invariance conjecture for the BCOV-type invariant of involutions on irreducible holomorphic symplectic fourfolds

Let $L_2$ be the Beauville–Bogomolov lattice of a manifold of $K3^{[2]}$-type. Let $M$ be an admissible sublattice of $L_2$, let $\mathcal{K}_1,\mathcal{K}_2\in\operatorname{KT}(M)$ be Kähler-type chambers, and let $(X_1,\iota_1)$ and $(X_2,\iota_2)$ be manifolds of $K3^{[2]}$-type with involution of types $(M,\mathcal{K}_1)$ and $(M,\mathcal{K}_2)$, respectively. Birational invariance conjecture. If there exists a $\mu_2$-equivariant birational map between $(X_1,\iota_1)$ and $(X_2,\iota_2)$, then

$$\tau_{M,\mathcal{K}_1}(X_1,\iota_1)=\tau_{M,\mathcal{K}_2}(X_2,\iota_2).$$

This is expected from the birational invariance of the BCOV invariant for Calabi–Yau manifolds and the preceding result in the paper; the asserted birational invariance of $\tau_{M,\mathcal{K}}$ is not established in the supplied context.