KPS Hecke-tree conjecture for rational branches

Let $P(t)\in\mathbb{F}_q[t]$ be irreducible. For a lattice $\Lambda_0\in\mathcal{L}_2$, let $\mathbb{T}_P(\Lambda_0)$ be its $P(t)$-Hecke tree, and let $\{\Lambda_n\}_{n\in\mathbb{N}}$ be a rational branch in this tree. Let $A$ denote the diagonal group acting on the space of lattices, and let $\mu_{A\Lambda_n}$ be the associated orbit measure. KPS's Hecke-tree conjecture. For every choice of rational branch $\{\Lambda_n\}_{n\in\mathbb{N}}$,

$$\lim_{n\rightarrow\infty}\mu_{A\Lambda_n}=0.$$

The source states this as a reformulation of the full escape-of-mass conjecture. Its parser status is unknown; the surrounding discussion establishes related results and says that the global structure of the sequence is known, but does not plainly resolve this assertion.