Conjecture on associated fields and tensorial transformation

Let $M$ be the underlying manifold, and let $D'^{m}_{n (A' o)}(M)$ be the indicated class of tensor distributions. An associated field means the field associated to such an object in the paper's sense. Transformation conjecture. If

$$T^{\mu\ldots}_{\nu\ldots}\in D'^{m}_{n (A' o)}(M)$$

has an associated field, then that associated field transforms on its domains as a tensor field. This would extend the tensorial behavior of associated fields beyond the already established cases, but the supplied text gives no resolution.