Let $C \subseteq \mathbb{R}^n$ and $D \subseteq \mathbb{R}^m$ be convex cones in Euclidean $n$- and $m$-space respectively, both containing their respective origins.
Let $M \in C \otimes D$ be given, i.e. $M = c_1 \otimes_{\mathbb{R}} d_1 + \cdots + c_N \otimes_{\mathbb{R}} d_N \in \mathbb{R}^n \otimes_{\mathbb{R}} \mathbb{R}^m$ for some $c_1, \dots,c_N \in C$ and $d_1, \dots, d_N \in D$.
If $M$ lies in the interior of $C \otimes D \subseteq \mathbb{R}^n \otimes_{\mathbb{R}} \mathbb{R}^m$, does there exists $p_1, \dots, p_M$ in the interior of $C$ and $q_1, \dots, q_M$ in the interior of $D$ such that $$M = p_1 \otimes_{\mathbb{R}} q_1 + \cdots + p_M \otimes_{\mathbb{R}} q_M?$$
Denoting the interior of a cone $C$ by $C^\circ$, the question is essentially asking whether $$(C \otimes D)^\circ = C^\circ \otimes D^\circ.$$