Yes, my example is easy to modify after some thought, just take a thick enough layer for each level.
More precisely, let $f$ be a sufficiently fast growing function, and define the initial value on any vertex at distance $f(n)\le d< f(n+1)$ from the root as $(-1)^n$.
Given $f(n)$, one can also pick a large enough $f(n+1)$ such that independently of the later values the root will get $(-1)^n$ in some time that depends only on $f(n+1)$ with at least $50\%$ probability.
This guarantees that with $1$ probability it will switch values infinitely often (just like every other vertex).