Finite-dimensional homology conjecture for degenerate minimal models

Let $p,q$ be positive integers, set

$$c_{p,q}=\frac{3}{2}\left(1-2\frac{(p-q)^2}{pq}\right),$$

and for integers $m,n$ set

$$h_{p,q}^{m,n}=\frac{(np-mq)^2-(p-q)^2}{8pq}.$$

For the Lie superalgebra ${\mathcal L}_s$ and the irreducible module $L(c_{p,q},h_{p,q}^{m,n})$, finite-dimensional homology conjecture.

$$\dim H_q({\mathcal L}_s,L(c_{p,q},h_{p,q}^{m,n}))<\infty$$

for every $q\in{\bf N}$. This predicts finite-dimensional homology in the minimal-model case, contrasting with the infinite-dimensional homology observed in the special non-minimal case discussed immediately before; the supplied text gives no resolution of the conjecture.