On the rectifiability of CD(K,N) and MCP(K,N) spaces with unique tangents

We prove rectifiability results for CD(K,N) and MCP(K,N) metric measure spaces (X,d,m) with pointwise Ahlfors regular reference measure m and with m-almost everywhere unique metric tangents. In particular, we show rectifiability if (i) (X,d,m) is CD(K,N) for an arbitrary N and has Hausdorff dimension $n<5$, or (ii) (X,d,m) is MCP(K,N) and non-collapsed, namely it has Hausdorff dimension N. Our strategy is based on the failure of the CD condition in sub-Finsler Carnot groups, on a new result on the failure of the non-collapsed MCP on sub-Finsler Carnot groups, and on the recent breakthrough by Bate [Invent. Math., 230(3):995-1070, 2022].