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Using tools and results from geometric measure theory, we give a simple newproof of the main result (Theorem 1.3) in K. Kondo and M. Tanaka, Approximationof Lipschitz Maps via Immersions and Differentiable Exotic Sphere Theorems,\textit{Nonlinear Anal.} \textbf{155} (2017), 219--249, as well as the conversestatement. It explores the connections between the theory of non-smoothanalysis {\it \`{a} la} F.~H. Clarke and the existence of special systems ofWhitney flat $1$-forms with Sobolev regularity on certain families of homologymanifolds.

Cartan--Whitney Presentation, Non-smooth Analysis and Smoothability of Manifolds: On a theorem of Kondo--Tanaka