State-counting formula for invariant soliton sectors

Let $m=(m^{(a)}_j)$ be a content such that

$${\mathcal T}(P(m))=P(m),$$

where

$${\mathcal T}(P(m))=\bigcup_{a=1}^n\bigcup_{j\geq1}\{T^{(a)}_j(p)\mid p\in P(m)\}.$$

Let $\Omega(m)$ be the Bethe-ansatz multiplicity, let $W$ be the Weyl group of $A_n$, and let $\lambda(m)$ be the associated dominant weight. State-counting conjecture. One has

$$\Omega(m)=\frac{|P(m)|}{|W\lambda(m)|}.$$

This relates the Bethe-ansatz count of solutions with content $m$ to the number of cellular-automaton states in the invariant sector, normalized by the size of the Weyl orbit of the associated weight. The supplied text gives no resolution status.