Conjecture on equivalence and iterated covariant derivatives

Let $M$ be the underlying manifold, let $U^i$ be a piecewise smooth tensor field, let the connection belong to $\Gamma^{3}_{S}(M)$, and let $T^{\mu\ldots}_{\nu\ldots},A^{\mu\ldots}_{\nu\ldots}\in\Gamma^{m}_{A(\cup_n A'_n\,o)}(M)$ in the indicated classes. Let $D^{n}_{C(U)}$ denote the $n$th covariant derivative along $U$. Derivative-preservation conjecture. If

$$T^{\mu\ldots}_{\nu\ldots}\approx A^{\mu\ldots}_{\nu\ldots},$$

then

$$D^{n}_{C(U)}T\approx D^{n}_{C(U)}A$$

for every natural number $n$, whenever such a covariant derivative exists. This proposes that equivalence is preserved by arbitrary iterated covariant differentiation under the stated existence condition; no resolution is supplied.