The transcendental-lattice conjecture for Nikulin orbifolds and fixed K3 surfaces

Let $X$ be a fourfold of $K3^{[2]}$-type admitting a symplectic involution $\sigma$, let $Y$ be the partial resolution of $X/\sigma$, and let $S$ be the K3 surface contained in $\operatorname{Fix}_\sigma(X)$. Denote by $T_Y$ and $T_S$ their transcendental lattices. Transcendental-lattice conjecture. One has

$$T_Y\simeq T_S.$$

This conjecture would upgrade the rational isometry established earlier in the paper to an integral isometry. The surrounding computations verify the expected relationship in the explicitly studied families, but the general assertion remains open in the source.