The orientation-preserving Hodge-isometry conjecture for K3 autoequivalences

Let $X$ be a K3 surface. Let

$$\Pi:\operatorname{Aut}^{\mathrm{FM}}\bigl(D^b\operatorname{Coh}(X)\bigr)\longrightarrow \operatorname{O}_{\mathrm{Hodge}}\bigl(\widetilde H(X,\mathbb Z)\bigr)$$

be the homomorphism induced by the action of Fourier--Mukai autoequivalences on the Mukai lattice, and let $\operatorname{O}_+\bigl(\widetilde H(X,\mathbb Z)\bigr)$ denote the subgroup of orientation-preserving Hodge isometries. The orientation-preserving Hodge-isometry conjecture. The image of $\Pi$ coincides with this subgroup:

$$\operatorname{im}(\Pi)=\operatorname{O}_+\bigl(\widetilde H(X,\mathbb Z)\bigr)\subseteq\operatorname{O}_{\mathrm{Hodge}}\bigl(\widetilde H(X,\mathbb Z)\bigr).$$

This conjecture seeks a cohomological description of Fourier--Mukai autoequivalences of K3 surfaces. The source presents it as an expectation and gives no resolution status.