Smoothness-boundary conjecture for Coble type quartics

Let $\Sigma^\circ\subset |\mathcal O_{\mathbb P^9}(4)|$ be the locus of Coble type quartics singular exactly along a smooth hyperkähler fourfold in $\mathcal M_4^{(1)}$, and let $\Sigma$ be its closure. Let $\Sigma_{16}$ and $\Sigma_{20}$ denote the proposed irreducible boundary divisors, and let $\Delta_{16}$ and $\Delta_{20}$ be the proposed $\operatorname{SL}(10)$ invariants. Smoothness-boundary conjecture. The complement $\Sigma\smallsetminus \Sigma^\circ$ is a reducible divisor, more precisely $\Sigma\smallsetminus \Sigma^\circ=\Sigma_{16}\cup \Sigma_{20}$ with both components irreducible. Equivalently, for $[f_4]\in \Sigma$, the singular locus of $Z(f_4)$ is a smooth hyperkähler fourfold in $\mathcal M_4^{(1)}$ if and only if $\Delta_{16}(f_4)\ne0$ and $\Delta_{20}(f_4)\ne0$. These divisors are presented as analogues of the divisor of singular cubic fourfolds, and the conjecture identifies the boundary of the smooth hyperkähler locus.