The degenerate singular-locus conjecture for Verra threefolds

Let $(J(T),\Xi)$ be the intermediate Jacobian of a general Verra threefold $T$, let $\Xi$ be its theta divisor, and let $e$ be the canonical extension class; write $\Xi_e$ for the corresponding translate. Degenerate singular-locus conjecture. The singular locus of the intersection $\Xi\cdot\Xi_e$ has dimension $4$ and has a unique component of that dimension. This component is a translate of

$$S_1^{\rm odd}+S_2^{\rm odd}.$$

This is the nodal degeneration of the preceding conjecture: the two period-map fiber components degenerate to the special surfaces $S_1^{\rm odd}$ and $S_2^{\rm odd}$. The source gives no resolution of this degenerate version.