The minimal-class conjecture for subvarieties of principally polarized abelian varieties

Let $(X,\Theta)$ be an irreducible principally polarized abelian variety, let $Y\subset X$ be a non-degenerate subvariety of dimension $m$, and write $\theta=c_1(\Theta)$. Say that $Y$ represents a minimal class when

$$[Y]=\frac{\theta^{g-m}}{(g-m)!}$$

in the appropriate cohomology group, equivalently in the notation of the source as the minimal class $\theta^m/m!$ according to its codimension convention. Say that $\mathcal I_Y(2\Theta)$ satisfies I.T. with index $0$ when its higher cohomology vanishes after general topologically trivial twists. Minimal-class conjecture. The subvariety $Y$ represents a minimal class if and only if $\mathcal I_Y(2\Theta)$ satisfies I.T. with index $0$, equivalently, $\mathcal I_Y$ is strongly $3$-$\Theta$-regular. This is presented as an analogue of the characterization of varieties of minimal degree in projective space; the source does not state whether the conjecture has been proved or disproved.