Collapsing and associated-variety conjecture for level $-6$ affine $F_4$ W-algebras

Let $X_V$ denote the associated variety of a vertex algebra $V$, and let $\mathbb{O}_{F_4(a_3)}$ be the nilpotent orbit of Bala–Carter type $F_4(a_3)$ in $F_4$. Level $-6$ $F_4$ conjecture.

$$\mathscr{W}_{-6}(F_4,F_4(a_3))\cong\mathbb{C},\qquad \mathscr{W}_{-6}(F_4,C_3(a_1))\cong L_{-1/2}(A_1),$$

and

$$X_{L_{-6}(F_4)}=\overline{\mathbb{O}_{F_4(a_3)}}.$$

The conjecture concerns the two undetermined collapsing cases at level $-6$ and the associated variety of the affine $F_4$ vertex algebra; the source later relates it to assumed instances of the Kac–Wakimoto conjecture.