The conjecture on loop-model modules with through-lines

Let $D4A2\mathscr W_{j,1}$ be the affine Temperley–Lieb loop-model module with $2j$ through-lines, let $\mathsf{V}_{0,-j}$ and $\overline{\mathsf{V}}_{0,j}$ be the corresponding left- and right-moving Verma modules, and let $\mathscr L_{e,j}$ denote the additional scaling-limit modules. For $j>0$, the proposed scaling limit is

$$\mathscr W_{j,1}\mapsto\left(\mathsf{V}_{0,-j}\otimes\overline{\mathsf{V}}_{0,j}\right)\oplus\bigoplus_{e=1}^{\infty}\mathscr L_{e,j}.$$

Loop-model modules with through-lines conjecture. For $j>0$ and $2j$ through-lines, the scaling limit of $\mathscr W_{j,1}$ is the direct sum displayed above. This extends the proposed Virasoro-module description from the sector without through-lines to sectors carrying through-lines and is part of the conjectural continuum decomposition of the generic-$Q$ loop model. The supplied text does not state whether this decomposition has been proved, so its resolution remains open here.