The unequal-parameter Kostant conjecture for multiples of the Weyl vector

Let $\mathfrak{g}$ be a simple Lie algebra or an untwisted affine Kac–Moody Lie algebra, let $\rho$ denote the sum of the fundamental weights, and let $V(\gamma)$ be the irreducible highest weight representation of highest weight $\gamma$. Let $P^+$ be the set of dominant integral weights, and take integers $m\geq n\geq0$. The unequal-parameter Kostant conjecture. If $\lambda\in P^+$ and

$$\lambda=m\rho+\beta$$

for some weight $\beta$ of $V(n\rho)$, then

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

This generalizes the equal-parameter case, replacing the sufficient condition $\lambda\leq2n\rho$ by a weight condition adapted to the second tensor factor. The paper proves the assertion after applying a saturation factor in general, and without one for type $A$ and for $\widehat{\mathfrak{sl}}_2$; the full conjecture remains open.