Associated-variety conjecture for the level $-12+9/7$ affine $E_6$ vertex algebra

Let $X_V$ denote the associated variety of a vertex algebra $V$, and let $\mathbb{O}_{E_6(a_1)}$ be the nilpotent orbit of Bala–Carter type $E_6(a_1)$ in $E_6$. Associated-variety conjecture.

$$X_{L_{-12+9/7}(E_6)}=\overline{\mathbb{O}_{E_6(a_1)}}.$$

The conjecture is motivated by the collapsing identity for the corresponding W-algebra and by the expected intersection with the relevant Slodowy slice; the level is non-admissible.