Conjectural deformation of secant sheaves on abelian sixfolds

Let $X$ be an abelian threefold, let $\hat X$ be its dual, and let ${{{{{{{{\mathcal G}}}}}}}}_{F_1,F_2}$ be the sheaf constructed from non-zero coherent sheaves $F_1,F_2$ whose Chern characters lie in a common $K$-secant plane. Let $\Theta_w\in\wedge^2V^*$ and $\hat J_{ww}\in\wedge^6_+V^*$ be the indicated $\operatorname{Spin}(V)_w$-invariant classes, where $w=ch(F_1)$. The secant-sheaf conjecture. The sheaf ${{{{{{{{\mathcal G}}}}}}}}_{F_1,F_2}$ is supported on a subscheme of codimension $1$ in $X\times\hat X$, all classes $ch_i({{{{{{{{\mathcal G}}}}}}}}_{F_1,F_2})$ are $\operatorname{Spin}(V)_w$-invariant, and

$$ch_3({{{{{{{{\mathcal G}}}}}}}}_{F_1,F_2})=s\hat J_{ww}+t\Theta_w^3,$$

with $s,t\in\mathbb Q$ and $s\ne0$; moreover, the pair $(X\times\hat X,{{{{{{{{\mathcal G}}}}}}}}_{F_1,F_2})$ deforms with $X\times\hat X$ to a pair $(A,{{{{{{{{\mathcal G}}}}}}}})$ for every abelian sixfold $A$ in the period domain $\Omega_w$. This predicts algebraic representatives for distinguished invariant Hodge classes throughout the relevant period domain, while the surrounding results use derived equivalences and deformation theory to establish related cases.