Boundary-level rationality and $C_2$-cofiniteness conjecture for affine $\mathfrak{g}$

Let $\mathfrak{g}=sl(2|1)$ and let $\widehat{\mathfrak{g}}$ be its affine Lie superalgebra. Let $m\in\mathbb{N}$ and set

$$\mathcal{k}=-\frac{m}{m+1}.$$

Let $L_{\widehat{\mathfrak{g}}}(\mathcal{k},0)$ be the simple affine vertex operator superalgebra at this boundary admissible level, let $\mathcal{O}$ be the category of modules under consideration, and let $\omega_\xi$ be the Virasoro element defining the associated $\mathbb{Q}$-graded vertex operator superalgebra.

Boundary-level rationality conjecture. The vertex operator superalgebra $L_{\widehat{\mathfrak{g}}}(\mathcal{k},0)$ is rational in the category $\mathcal{O}$, and its irreducible weak modules in $\mathcal{O}$ are exactly the admissible modules of level $\mathcal{k}$ for $\widehat{\mathfrak{g}}$. Moreover, the $\mathbb{Q}$-graded vertex operator superalgebra $(L_{\widehat{\mathfrak{g}}}(\mathcal{k},0),\omega_\xi)$ is rational and $C_2$-cofinite.

The paper establishes the corresponding assertions at the boundary level $-\frac12$ and gives non-boundary examples where rationality fails. The general statement for the boundary levels $\mathcal{k}=-\frac{m}{m+1}$ remains open.