The tensor-power occurrence conjecture for semisimple Lie algebras

Let $\mathfrak{g}$ be a semisimple Lie algebra and let $V(\lambda)$ be an irreducible $\mathfrak{g}$-module of dimension $r$, with highest weight $\lambda$. Let $\Pi(\lambda)$ denote the weights of $V(\lambda)$, let $\mu\preceq\lambda$ be a dominant weight in $\Pi(\lambda)$, and let $\pi=\Lambda/\Lambda_r$ be the fundamental group, where $\Lambda$ is the weight lattice and $\Lambda_r$ the root lattice. Tensor-power occurrence conjecture. The tensor power $V(\lambda)^{\otimes n}$ contains the irreducible $\mathfrak{g}$-module $V(\mu)$ for some $n\leq\max_{g\in\pi}\operatorname{ord}(g)$ if $\pi$ is nontrivial, and for $n=2$ otherwise. The conjecture proposes an upper bound, matching the preceding lower-bound discussion, for the tensor power in which a prescribed constituent first occurs. The source gives no evidence of resolution.