Drinfeld-Sokolov reduction conjecture for simple modules of exceptional W-algebras

Let $\mathfrak{g}=\mathfrak{sp}_4$, let $f$ be a subregular nilpotent element, and let $k$ be an admissible level. Write $L(\xi^{(s)}_{i,j},\chi^{(s)}_{i,j})$ for the simple $\mathcal{W}_k(\mathfrak{g},f)$-modules specified in Proposition 2.7. Drinfeld-Sokolov reduction conjecture. All these simple modules are obtained by Drinfeld-Sokolov reduction of highest-weight $\widehat{\mathfrak{g}}$-modules. More precisely, if $k=-3+p/3$ with ${((,p)},3)=1$ and $p\geqslant 3$, then

$$H^l_f\left(\widehat{L}_k(\lambda^{(s)}_{i,j})\right)= \begin{cases} L(\xi^{(s)}_{i,j},\chi^{(s)}_{i,j}) & \text{if } l=0,\\ 0 & \text{otherwise}, \end{cases}$$

for the parameters in Proposition 2.7P; and if $k=-3+p/4$ with ${((,p)},2)=1$ and $p\geqslant 4$, the same formula holds for the parameters in Proposition 2.7CP. This conjecture would identify every listed simple W-algebra module with a degree-zero Drinfeld-Sokolov reduction and assert vanishing in all other degrees. The statement is presented after explicit character computations for several families, but the source provides no proof of the full claim.