The structure conjecture 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, and let ${\mathcal P}(G)$ be the tensor product cone. For a triple of dominant weights $\sigma=(\lambda_1,\lambda_2,\lambda_3)$, write $Tr(\sigma)$ for its total weight and call the triple nonsingular if none of its components lies on a wall of the Weyl chamber.

The structure conjecture. (1) If $R$ is simply-laced, then

$$Tens(G)=\{\sigma=(\lambda_1,\lambda_2,\lambda_3)\in {\mathcal P}(G): \lambda_i\in L,\ Tr(\sigma)\in Q(R)\}.$$

(2) If $\sigma=(\lambda_1,\lambda_2,\lambda_3)\in {\mathcal P}(G)\cap L^3$ is nonsingular, then $\sigma\in Tens(G)$ if and only if $\sigma\in {\mathcal P}(G)$ and $Tr(\sigma)\in Q(R)$. (3) If $P(R)=Q(R)$, then in the decomposition of $Tens(G)$ as the union of elementary sets, the elementary sets are given by inequalities only and there are no congruence conditions.

The conjecture holds for the root systems $A_n$, $B_2=C_2$, and $G_2$, and is supported by computer calculations for higher-rank root systems. A related observation is that for nonsingular triples, the only congruence condition is the necessary condition $Tr(\sigma)\in Q(R)$; the general assertions remain open.