Frequently dense harmonic functions and universal martingales on trees

We prove the existence of harmonic functions $f$ on trees, with respect to suitable transient transition operators $P$, that satisfy an analogue of Menshov universal property in the following sense: $f$ is the Poisson transform of a martingale on the boundary of the tree (equipped with the harmonic measure $m$ induced by $P$) such that, for every measurable function $h$ on the boundary, it contains a subsequence that converges to $h$ in measure. Moreover, the martingale visits every open set of measurable functions with positive lower density.