Rationality conjecture for non-admissible minimal W-algebras

Let $\mathfrak{g}$ be a simple Lie algebra of type $d_4$, $e_6$, $e_7$, or $e_8$, and let $m\geq -1$ be an integer. Set

$$k(m)=-\frac{h^{\vee}}{6}+m,$$

where $h^{\vee}$ is the dual Coxeter number of $\mathfrak{g}$, and let $\mathscr{W}_{k(m)}(\mathfrak{g},f_\theta)$ denote the simple minimal W-algebra. Rationality conjecture for minimal W-algebras. The algebra $\mathscr{W}_{k(m)}(\mathfrak{g},f_\theta)$ is rational if and only if $k(m)<0$; equivalently, $m=-1,0$ for $d_4$, $m=-1,0,1$ for $e_6$, $m=-1,0,1,2$ for $e_7$, and $m=-1,0,1,2,3,4$ for $e_8$. These are lisse non-admissible W-algebras, and the conjecture predicts that the remaining non-negative levels give a new family of lisse logarithmic vertex algebras. The source supplies no resolution of the full assertion.