Conjecture on preservation of equivalence under tensor multiplication

Let $M$ be the underlying manifold, and let $C,D,F,B\in\Gamma^{m}_{A}(M)$ belong to the same indicated classes $\Gamma^{m}_{A(At(\tilde A))}(M)$. Assume that, for every relevant $\tilde A$ and every $A'\subseteq\tilde A$, all four objects have associated fields defined on the whole of $M$ for every $At(\tilde A)$. Equivalence–multiplication conjecture. If

$$F\approx B\quad\text{and}\quad C\approx D,$$

then

$$C\otimes F\approx D\otimes B.$$

This proposes that the paper's equivalence relation is stable under tensor multiplication under the stated atlas and association hypotheses; no resolution is supplied.