The conjecture on the quotient loop-model module without through-lines

Let $D4A2\mathscr W_{0,\mathfrak q^{\pm 2}}$ denote the quotient loop-model module without through-lines, and let $\mathsf{X}_{n,1}$ and $\overline{\mathsf{X}}_{n,1}$ be irreducible left- and right-moving Virasoro modules, respectively. In the scaling limit,

$$\overline{\mathscr W}_{0,\mathfrak q^{\pm 2}}\mapsto\bigoplus_{n=1}^{\infty}\mathsf{X}_{n,1}\otimes\overline{\mathsf{X}}_{n,1}.$$

Quotient loop-model module without through-lines conjecture. For generic $\mathfrak q$, the scaling limit of the quotient loop-model module without through-lines is the direct sum displayed above. This concerns the Virasoro-module decomposition of the $j=0$ sector and does not involve Jordan blocks. The modules $\mathsf{X}_{r,s}$ are irreducible Virasoro modules with highest weight $h_{r,s}$; in this case they are Kac modules with the submodule generated by the unique singular vector at level $rs$ factored out. The result was reported as well checked, but its general status is not otherwise established in the supplied text.