Fourier–Mukai equivalence of conjectural stability hearts on abelian varieties

Let $X$ and $Y$ be derived equivalent $g$-dimensional abelian varieties, with Fourier–Mukai transform

$$\Phi_{\mathcal{E}}^{X \to Y}:D^b(X)\to D^b(Y).$$

Let $D_X$ and $D_Y$ be the divisor classes associated to the equivalence, and let $\ell_X\in\operatorname{NS}_{\mathbb{Q}}(X)$ and $\ell_Y\in\operatorname{NS}_{\mathbb{Q}}(Y)$ be the induced ample classes. For $k\in\{1,2,\ldots,g-1\}$ and $\lambda\in\mathbb{R}_{>0}$, set

$$\Omega=-D_X+\lambda e^{ik\pi/g}\ell_X,\qquad \Omega'=D_Y-(1/\lambda)e^{-ik\pi/g}\ell_Y.$$

Fourier–Mukai heart-equivalence conjecture. The Fourier–Mukai transform gives an equivalence of the conjecturally constructed stability-condition hearts:

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

This predicts that derived equivalences between abelian varieties preserve the recursively constructed hearts underlying their conjectural Bridgeland stability conditions, after the indicated shift and transformation of complexified ample classes. It is proposed for all abelian varieties; the relevant stability conditions are known in dimensions two and three, while the general-dimensional case remains conjectural.