Rational Hodge isometries conjecture for hyper-Kähler varieties of K3[n]-type

Let ${\mathcal G}^{[n]}_{an}$ be the groupoid of the relevant cohomological morphisms between projective irreducible holomorphic symplectic manifolds of $K3^{[n]}$-type, and let ${\mathcal R}^{[n]}_{an}$ be the subgroupoid consisting of morphisms whose normalization by $\mu_{\chi(\phi)}$ conjugates the cup product to itself in the degree-preserving case and to the Pontryagin product $\star$ in the degree-reversing case. For a Fourier–Mukai kernel $F\in D^b(X\times Y)$, write $\kappa(F)$ for its Mukai-type class and let $\mu_t$ multiply $H^{2k}(X,\mathbb Q)[\dim(X)]$ by $t^k$. Rational Hodge isometries conjecture. The subgroupoid ${\mathcal R}^{[n]}_{an}$ is the whole of ${\mathcal G}^{[n]}_{an}$. Furthermore, if an object of positive rank $F$ in $D^b(X\times Y)$ is the Fourier–Mukai kernel of an equivalence $\Phi_F:D^b(X)\rightarrow D^b(Y)$ between projective irreducible holomorphic symplectic manifolds $X$ and $Y$ of $K3^{[n]}$-type, then

$$\operatorname{rank}(F)=n!t^n$$

for some rational number $t$, and, setting

$$\phi:=\mu_t\circ [\kappa(F)\sqrt{\operatorname{td}_{X\times Y}}]_*:H^*(X,\mathbb Q)\rightarrow H^*(Y,\mathbb Q),$$

we have

$$\phi(\gamma_1\cup\gamma_2)=\phi(\gamma_1)\star\phi(\gamma_2)$$

for all $\gamma_1,\gamma_2\in H^*(X,\mathbb Q)$. This extends the expected compatibility between normalized cohomological Fourier–Mukai transforms and the cup and Pontryagin products; the supplied text does not establish whether the claim has been resolved.