The singular-triple criterion for tensor product semigroups

Let $G$ be a complex semisimple Lie group with root system $R$, let $L$ be the character lattice of a maximal torus, let ${\mathcal P}(G)$ be the tensor product cone, and let $Tr(\sigma)$ denote the total weight of a triple. A triple of dominant weights is singular if at least one component lies on a wall of the Weyl chamber.

Singular-triple criterion. There exists a triple $\sigma\in {\mathcal P}(G)\cap L^3$ with $Tr(\sigma)\in Q(R)$ and $\sigma\notin Tens(G)$ if and only if there exists a singular triple with these same properties.

This is presented as a less ambitious consequence of either Part 1 or Part 2 of the preceding structure conjecture. The source gives no resolution beyond the stated relationship, so the criterion remains open in general.