Median-preserving conjecture for the stopped-random-walk tree dynamics

Let $\{L,R\}^{\star}$ be the set of finite words over $\{L,R\}$, and let $P_v$ denote the probability measure on $\mathbb{Z}$ attached to the tree node indexed by $v$. For a word $v$, write $P_v(\mathbb{N})$ and $P_v(-\mathbb{N})$ for the masses on the nonnegative and nonpositive integers, respectively.

Tree-asymmetry conjecture. For every $v\in\{L,R\}^{\star}$,

$$P_{Lv}(\mathbb{N})>\frac{1}{2}\quad\text{and}\quad P_{Rv}(-\mathbb{N})>\frac{1}{2}.$$

This asserts that the strict asymmetry created by the first left or right step persists throughout the entire tree evolution. It is presented as a generalization of Cusick's conjecture; the weaker median inequalities are discussed separately, and the proposed strict statement remains open.