The corrected saturation conjecture for semisimple Lie groups

Let $G$ be a connected complex semisimple Lie group with maximal torus $T$. Let $(\lambda,\mu,\nu)$ be a triple of dominant weights for $G$, and let $N$ be a positive number such that

$$\left(V_{N\lambda}\otimes V_{N\mu}\otimes V_{N\nu}\right)^G>0.$$

Saturation conjecture for semisimple Lie groups. If $\lambda+\mu+\nu$ annihilates all elements of $T$ with semisimple centralizer, then

$$\left(V_{\lambda}\otimes V_{\mu}\otimes V_{\nu}\right)^G>0.$$

The preceding naive saturation statement for arbitrary semisimple groups is explicitly reported to be false, while this corrected formulation is posed as the appropriate conjecture. The source does not state a resolution, so its status remains open here.