Consistency of fixed-point partitions for branching matrices

Let $\mathbf{M}$ be a branching matrix, and for a specific value $\lambda<\operatorname{WSM}(T_{\mathbf{M}})$ let $V_\lambda$ be the fixed points of the recurrences of the marginal distributions defined by $\mathbf{M}$. Define $\mathcal{C}(\lambda)$ to be the partition whose parts consist of types having the same value at the fixed points in $V_\lambda$; that is, for each $C_i\in\mathcal{C}(\lambda)$ and every $c\in C_i$, the values $V_\lambda(c)$ are equal. Fixed-point partition conjecture. If the partitions $\mathcal{C}(\lambda)$ are identical for all $\lambda$, then $\mathcal{C}$ is a consistent partition of $\mathbf{M}$. The conjecture would provide a way to find a small consistent partition and thereby simplify proofs of weak and strong spatial mixing for the associated trees. The source does not state a resolution, so the conjecture is recorded as open.