Simplicity conjecture for vacuum modules of positive-defect Lie superalgebras

Let $\mathfrak{g}$ be an (almost) simple finite-dimensional Lie superalgebra of positive defect, namely one of $\mathfrak{sl}(m,n)$ with $m,n\geq 1$, $\mathfrak{osp}(m,2n)$ with $m\geq 2$ and $n\geq 1$, $D(2,1,a)$, $F(4)$, or $G(3)$. For a non-isotropic root $\alpha$, define

$$k_{\alpha}:=\frac{k+h^{\vee}_{B}}{B(\alpha|\alpha)}.$$

Here $V^k$ denotes the vacuum module for the affine Lie superalgebra $\widehat{\mathfrak{g}}$. Vacuum-module simplicity conjecture. The $\widehat{\mathfrak{g}}$-module $V^k$ is not irreducible if and only if

$$k_{\alpha}\in\mathbb{Q}_{\geq 0}$$

for some even root $\alpha$ of $\mathfrak{g}$. This conjecturally characterizes reducibility of vacuum modules for the listed positive-defect Lie superalgebras; the supplied text does not state whether the claim has been proved or disproved.