Pairwise locality compatibility of bilinear relations

Let $(E,\top)$ be a locality vector space, and let $V_1,V_2$ be subspaces of $E$. Write $I_{\rm bil}(V_1,V_2)$ for the subspace of bilinear relations and $\mathbb{K}(V_1\times_\top V_2)$ for the linear span of the locality-compatible pairs. Pairwise locality compatibility conjecture. The subspace

$$I_{\rm bil}(V_1,V_2)\cap\mathbb{K}(V_1\times_\top V_2)$$

is locality compatible with $\top_{V_1\times_\top V_2}$. This would ensure that the locality tensor product of two subspaces inherits a locality vector-space structure; the paper uses this as the basic conjectural condition for extending the result to tensor products of several subspaces.