Associated-variety conjecture for the level $-18+14/5$ affine $E_7$ vertex algebra

Let $X_V$ denote the associated variety of a vertex algebra $V$, and let $\mathbb{O}_{E_7(a_5)}$ be the nilpotent orbit of Bala–Carter type $E_7(a_5)$ in $E_7$. Associated-variety conjecture.

$$X_{L_{-18+14/5}(E_7)}=\overline{\mathbb{O}_{E_7(a_5)}}.$$

The source motivates this by analogy with the $D_5$ case and by a claimed finite extension of an admissible affine vertex algebra; the conjecture remains unproved in the supplied text.