Rectifiability of planes and Alberti representations

We study metric measure spaces that have quantitative topological control, as well as a weak form of differentiable structure. In particular, let $X$ be a pointwise doubling metric measure space. Let $U$ be a Borel subset on which the blowups of $X$ are topological planes. We show that $U$ can admit at most $2$ independent Alberti representations. Furthermore, if $U$ admits $2$ Alberti representations, then the restriction of the measure to $U$ is $2$-rectifiable. This is a partial answer to the case $n=2$ of a question of the second author and Schioppa.