The logarithmic module-count conjecture for the triplet vertex algebra

Let $p,p'\in\mathbb{Z}_{>0}$ with $p\ne p'$ and $p\geq 2$. Let $L=\mathbb{Z}\alpha$ satisfy $\langle\alpha,\alpha\rangle=2pp'$, and let $\mathcal{W}(p,p')$ be the vertex operator algebra defined as the intersection of the kernels of the two screening operators associated with this lattice construction. The logarithmic module-count conjecture. The vertex algebra $\mathcal{W}(p,p')$ is $C_2$-cofinite and has $2pp'+\frac{(p-1)(p'-1)}{2}$ inequivalent irreducible modules. This conjecture concerns the finiteness and representation theory of the logarithmic minimal-model-type vertex algebra; the supplied text does not state whether it has been resolved.