Affine Kostant tensor product conjecture for untwisted affine Kac–Moody algebras

Let $\mathfrak{g}$ be an untwisted affine Kac–Moody Lie algebra, let $\rho$ be the affine Weyl weight used in the tensor product $V(\rho)\otimes V(\rho)$, and let $\lambda\in\mathcal{P}^+$ be a dominant integral weight. Write $\lambda\leq2\rho$ in the dominance order. Affine Kostant conjecture. If $\lambda\leq2\rho$, then

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

The theorem in the paper proves this assertion for type $A_n^{(1)}$, and computations support it in several low-rank additional types. The assertion remains open for arbitrary untwisted affine Kac–Moody algebras.