Conjecture on covariant differentiation for piecewise smooth data

Let $U^i$ be a piecewise smooth vector field and let $\Gamma^{\mu}_{\nu\alpha}\in\Gamma^{3}_{S}(M)$. Piecewise-smooth covariant-derivative conjecture. Covariant differentiation along $U^i$ defines a map

$$D'^{m}_{n (\cup_n A'_n\, o)}(M)\to\tilde\Gamma^{m+n}_{A (\cup_n A'_n\,o)}(M),$$

and the image classes $\tilde\Gamma^{m+n}_{A (\cup_n A'_n\,o)}(M)$ contain exactly one element of $D'^{m}_{n}(M)$. This is a broader extension of the preceding covariant-derivative proposal to piecewise smooth vector fields and connections; no resolution is supplied.