Fourier–Mukai compatibility conjecture for iterated-tilt hearts

Let $X$ and $Y$ be abelian varieties of the same dimension $g$, let $\Phi_{\mathcal{E}}^{X\to Y}:D^b(X)\to D^b(Y)$ be a Fourier–Mukai transform, and let $\mathcal{A}^{X}_{\Omega_k}$ and $\mathcal{A}^{Y}_{\Omega'_k}$ denote the $g$-fold-tilt hearts associated with complexified ample classes $\Omega_k$ and $\Omega'_k$.

Fourier–Mukai compatibility conjecture. For $1\leq k\leq g-1$, there are suitable complexified ample classes $\Omega_k$ and $\Omega'_k$ such that

$$\Phi_{\mathcal{E}}^{X\to Y}[k](\mathcal{A}^{X}_{\Omega_k})=\mathcal{A}^{Y}_{\Omega'_k}.$$

This predicts that Fourier–Mukai equivalences identify the hearts produced by the iterated-tilt construction. It depends on the conjectural existence of the relevant stability conditions and remains open in the stated generality.