The boundary-stratum conjecture for tensor product semigroups

Let $G$ be a complex semisimple Lie group, let ${\mathcal P}(G)$ be its tensor product cone, let $\Lambda=\{\sigma\in L^3:Tr(\sigma)\in Q(R)\}$, and let $S$ be a stratum in the boundary of ${\mathcal P}(G)$ containing a point $(\lambda,\mu,\nu)$ whose three components are regular.

Boundary-stratum conjecture. There exists a subcone $S+\sigma\subset S$ such that

$$(S+\sigma)\cap\Lambda=(S+\sigma)\cap Tens(G).$$

Thus, sufficiently far inside such a boundary stratum, the lattice and root-lattice conditions are conjectured to be sufficient for membership in the tensor product semigroup. The source does not provide a resolution of this assertion in the supplied passage.