Jordan-cell rank conjecture for the vacuum tilting module

Let $\mathcal{T}_{0,\mathfrak{q}^2}$ be the vacuum tilting module in the scaling limit of periodic Jones–Temperley–Lieb modules, and let $\mathcal{X}_{j,P}$ denote an irreducible subquotient with labels $j$ and $P$. The Hamiltonian is $L_0+\bar L_0$.

Jordan-cell rank conjecture. In the scaling limit of $\mathcal{T}_{0,\mathfrak{q}^2}$, states from limits of irreducible $\mathcal{X}_{j,P}$ subquotients should be involved in Jordan cells of $L_0+\bar L_0$ of maximum rank

$$\operatorname{max\ rank}(L_0+\bar L_0\text{ in }\mathcal{T}_{0,\mathfrak{q}^2})=\begin{cases}3,&j=2,\ P=1,\\2\left\lceil\frac{j-2}{3}\right\rceil,&j\bmod 3=0\text{ and }P=\mathfrak{q}^{\pm2},\ j>0,\\2\left\lceil\frac{j-2}{3}\right\rceil+1,&j\bmod 3=1\text{ or }2\text{ and }j>2\text{ and }P=1. \end{cases}$$

This conjecture refines the lattice analysis by predicting that the maximum Jordan-cell rank increases by one in the scaling limit. The expected ranks include $3$ for states from $\mathcal{X}_{2,1}$, rank $2$ for $\mathcal{X}_{3,\mathfrak{q}^{\pm2}}$, rank $3$ for $\mathcal{X}_{4,1}$, and rank $4$ for $\mathcal{X}_{6,\mathfrak{q}^{\pm2}}$; the general statement remains open.