Grothendieck-ring conjecture for principal exceptional W-algebras

Let $n\geq 2$ and let $k=-h^\vee+\frac{p}{2n-1}$ be an exceptional level. Let $\mathcal{K}(L_{p-h^\vee}(\mathfrak{so}_{2n+1}))$ be the Grothendieck ring of the corresponding affine category, and let $\mathcal{K}^0(\mathcal{W}_k^{D^+}(n,1))$ be the fusion subring generated by the modules $\mathbf{L}_k(\lambda)$. Principal-exceptional Grothendieck-ring conjecture. There is an isomorphism

$$\mathcal{K}(L_{p-h^\vee}(\mathfrak{so}_{2n+1}))\xrightarrow{\simeq}\mathcal{K}^0(\mathcal{W}_k^{D^+}(n,1)),\qquad L_{p-h^\vee}(\lambda)\mapsto\mathbf{L}_k(\lambda).$$

The conjecture is motivated by the identification of integral admissible weights with dominant weights and by examples, but the source gives no resolution.