Singular-locus conjecture for secant varieties of spinor varieties

Let $N\geq 7$, let $\mathbb S_N^+$ be the Spinor variety, let $\sigma_2(\mathbb S_N^+)$ be its secant variety of lines, and let $\Sigma_{2,N}$ be the distance--$2$ orbit. The singular locus is denoted by $\operatorname{Sing}(\sigma_2(\mathbb S_N^+))$. Singular-locus conjecture. For any $N\geq 7$,

$$\operatorname{Sing}\left(\sigma_2(\mathbb S_N^+)\right)=\overline{\Sigma_{2,N}}=\mathbb S_N^+\sqcup\Sigma_{2,N}.$$

The preceding result gives the inclusions $\overline{\Sigma_{2,N}}\subseteq\operatorname{Sing}(\sigma_2(\mathbb S_N^+))\subseteq\tau(\mathbb S_N^+)$, so the conjecture identifies the singular locus exactly with the closure of the distance--$2$ orbit. It is motivated by the arguments in the appendix; the source provides no resolution of the conjecture.