Kostant's tensor-product conjecture for the Weyl vector

Let $\mathfrak{g}$ be a complex semisimple Lie algebra with weight lattice notation $P^+$, and let $\rho$ be the half-sum of its positive roots. For a dominant integral weight $\lambda\in P^+$, write $\lambda\leq 2\rho$ for the dominance order, and let $V(\lambda)$ denote the irreducible highest weight representation of highest weight $\lambda$. Kostant's conjecture. If $\lambda\leq 2\rho$, then

$$V(\lambda)\subseteq V(\rho)\otimes V(\rho).$$

The conjecture asserts that every dominant highest weight allowed by the dominance bound occurs in the tensor product. It is known for type $A$ and up to a saturation factor in general; it has also been checked for exceptional types, but remains open for the other classical simple Lie algebras.