Göpel's orbital degeneracy-locus conjecture for the half-spin representation

Let $\Delta$ be the half-spin representation, let $\mathbb{Q}_{14}$ be the $14$-dimensional quadric, and let $\mathcal S$ be the associated spinor bundle. For the codimension-$i$ orbit closures $Y_i$ in the fiber half-spin representation, define

$$D_{Y_i}(v)=\{x\in\mathbb{Q}_{14}\mid v(x)\in Y_i\subset\Delta_{14}\}.$$

Here $Qu_v=D_{Y_1}(v)$ is the quartic section associated with $v$, and $\operatorname{Jac}(C)/\{\pm\operatorname{id}\}$ denotes the Kummer variety of the curve $C$.

Göpel's orbital degeneracy-locus conjecture. Let $v$ be a general element of $\Delta$. Then:

1. $D_{Y_{10}}(v)$ is the Kummer fourfold $\operatorname{Jac}(C)/\{\pm\operatorname{id}\}$ of a non-hyperelliptic genus-four curve $C$ with a vanishing theta null and a unique degree-$3$ map to $\mathbb{P}^1$;
2. $D_{Y_{14}}(v)$ consists of the $256$ $2$-torsion points in the Kummer;
3. $D_{Y_5}(v)$ is the moduli space $\operatorname{SU}_C(2,\mathcal O_C)$ of semistable rank-$2$ vector bundles on $C$ with trivial determinant;
4. $D_{Y_1}(v)=Qu_v$ is the unique quartic singular along $\operatorname{SU}_C(2,\mathcal O_C)$.

The conjecture proposes a detailed correspondence between orbital degeneracy loci of a general half-spin section and moduli spaces associated with a special genus-four curve. The supplied text gives no evidence that it has been solved or refuted.