Stability conjecture for smooth 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 $\Sigma^s$ denote the stable locus for the action of $\operatorname{SL}(10)$. Stability conjecture. Every $[f_4]\in \Sigma^\circ$ is stable under the action of $\operatorname{SL}(10)$; equivalently, $\Sigma^\circ\subset \Sigma^s$. If true, this gives geometric quotient loci and supports the open immersion into $\mathcal M_4^{(1)}$ described subsequently.