Regularity via minors for odd k

Let $d>2$, and let $\Omega$ be an open subset of $\mathbb{R}^d$. Let $2 \le k \le d-1$ be a fixed odd integer and let $p \ge 1$. Let $f \in W^{1,p}(\Omega,\mathbb{R}^d)$. For a linear map $A:\mathbb{R}^d\to\mathbb{R}^d$, write $\bigwedge^k A$ for its induced map on $\bigwedge^k\mathbb{R}^d$, and define

$$\widetilde H_{>k}=\left\{B\in\operatorname{End}\left(\bigwedge^k\mathbb{R}^d\right)\mathrel{\middle|}\operatorname{rank}(B)>k\right\}.$$

Regularity via minors (odd $k$). If $\bigwedge^k df\in\widetilde H_{>k}$ is smooth, then $f$ is smooth.

This proposes extending the constant-rank regularity theorem to maps whose differential need not have constant rank. The preceding discussion notes that the exterior-power map is injective, and even an immersion, on the relevant rank locus; whether this analytic regularity implication holds is left unresolved here.