Coset Virasoro realization conjecture for the Fock spaces

Let

$$\mathcal{F}_{m,\ell}(n)=\bigoplus_{M=0\atop \mathrm{mod}\ 2k}^{2k-1}\mathcal{F}_{M;m,\ell,n},$$

where $\mathcal{F}_{M;m,\ell,n}$ are the Fock-space components in the decomposition of the $U_{x,p}(\widehat{\mathfrak{sl}}_2)$-module, and let $a$ satisfy $a\equiv m+\ell\pmod 2$. Coset Virasoro realization conjecture. The space $\mathcal{F}_{m,\ell}(n)$ is isomorphic to the irreducible coset Virasoro module $Vir_{m,a}$ with central charge

$$c_{Vir}=\frac{3k}{k+2}\left(1-\frac{2(k+2)}{4rr^*}\right)$$

and highest weight

$$h_{m,a}=\frac{\ell(k-\ell)}{2k(k+2)}+\frac{(mr-ar^*)^2-k^2}{4krr^*}.$$

For generic $r$, the preceding character comparison supports this identification; when $r$ is an integer greater than $k+2$, the source instead describes a BRST resolution yielding the irreducible coset Virasoro minimal module. The conjectural isomorphism is not resolved in the supplied text.