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.