The associated-variety conjecture for the simple vertex algebra $L_{-2}(G_2)$

Let $L_{-2}(G_2)$ be the simple vertex algebra at level $-2$, and let $X_{L_{-2}(G_2)}$ denote its associated variety. Let $\mathbb{O}_{\mathrm{subreg}}$ be the subregular nilpotent orbit of $G_2$, and write $\overline{\mathbb{O}}_{\mathrm{subreg}}$ for its closure. Associated-variety conjecture. The associated variety is

$$X_{L_{-2}(G_2)}=\overline{\mathbb{O}}_{\mathrm{subreg}}.$$

The conjecture would identify the geometric support of this simple vertex algebra with the subregular nilpotent orbit closure. The source presents it as a direction for future work and does not report a proof or disproof.