Cofiniteness conjecture for relative semi-infinite cohomology of admissible affine algebras

Let $L_k(\mathfrak{sl}_n)$ be the simple affine vertex algebra at the admissible level $k=-n+\frac{p}{q}$. Cofiniteness conjecture. The following vertex algebras are $C_2$-cofinite: (1) $\mathrm{H}_{\mathrm{rel}}^0(\mathfrak{h},L_k(\mathfrak{sl}_2)\otimes V_{\sqrt{q(2q-p)}A_1})$ for $2q>p\geq2$ and $q\geq2$; (2) $\mathrm{H}_{\mathrm{rel}}^0(\mathfrak{h},L_k(\mathfrak{sl}_3)\otimes V_{\sqrt{2(6-p)}A_2})$ for $6>p\geq3$ and $q=3$. The claim is a concrete consequence predicted from the relative semi-infinite cohomology conjecture; the source excerpt does not report a proof or disproof.