Critical-level conjecture for superelliptic affine Lie algebras

Let $r$ and $m$ be the parameters of a superelliptic algebra, and set

$$\nu = \frac{r}{2}\left(1+\frac{1}{m}\right).$$

Let $\mathfrak{g}$ be the underlying Lie algebra, let $h^{\vee}$ be its dual Coxeter number, and let $f$ be an explicit rational function. Critical-level conjecture. For each superelliptic algebra with parameters $(r,m)$, the critical level is

$$k_{\mathrm{crit}}(r,m) = -h^{\vee} + f\!\left(\frac{r}{2}\left(1+\frac{1}{m}\right)\right).$$

In particular, for the DJKM case $(r,m)=(1,2)$, this recovers the classical critical level $k_{\mathrm{crit}}=-h^{\vee}$. The conjecture concerns how the critical level depends on both superelliptic parameters in the vertex-algebra framework; the function $f$ is not specified in the source.