Conjecture on association commuting with tensor products

Let $M$ be the underlying manifold. Let $\Gamma^{m}_{(A' o)}(M)$ and $\Gamma^{n}_{S}(M)$ be the indicated classes, and let $A_s$ denote the association map. Association–tensor-product conjecture. If

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

has an associated field and

$$L^{\alpha\ldots}_{\beta\ldots}\in\Gamma^{n}_{S}(M),$$

then

$$A_s(T\otimes L)=A_s(T)\otimes A_s(L),$$

where the tensor product has slightly different meanings on the two sides. The claim proposes compatibility of association with tensor multiplication; the supplied text gives no resolution.