The reducibility criterion for modules over the twisted Heisenberg–Virasoro algebra

Let $p\in\mathbb{N}$, and let $c,h,h_W,r$ be the parameters of the module $L^{\prime}(c,h,h_W)$. Assume

$$2h_W+\frac{p^2-1}{12}c=0.$$

Reducibility criterion. The module $L^{\prime}(c,h,h_W)$ is reducible if and only if

$$h=h_W+\frac{(13p+1)(p-1)}{12}+\frac{(1-r)p}{2}.$$

This gives an explicit parameter criterion for reducibility in the stated case. The supplied text does not provide evidence that the criterion has been proved or resolved beyond its presentation as a conjectural statement.