Conjecture on tensor multiplication of generalized tensor distributions

Let $M$ be the underlying manifold, and let $\tilde D'^{a}_{b E A}(M)$ and $\tilde D'^{m}_{n E A}(M)$ denote the indicated classes of generalized tensor distributions, while $\tilde \Gamma^{a}_{E A}(M)$ and $\tilde \Gamma^{b}_{E A}(M)$ denote the indicated classes of generalized tensor fields. Tensor multiplication conjecture. Tensor multiplication gives the maps

$$\tilde D'^{a}_{b E A}(M)\times\tilde D'^{m}_{n E A}(M)\to \tilde D'^{a+m}_{b+n E A}(M)$$

and

$$\tilde \Gamma^{a}_{E A}(M)\times\tilde \Gamma^{b}_{E A}(M)\to \tilde \Gamma^{a+b}_{E A}(M).$$

This is one of the paper's proposed extensions of multiplication to suitable subclasses; the supplied text gives no resolution, and the notation and precise domains depend on the paper's generalized-distribution framework.