The affine δ-maximal component conjecture for unequal tensor parameters

Let $\mathfrak{g}$ be an untwisted affine Kac–Moody Lie algebra, let $\delta$ be its null root, and let $V(\gamma)$ denote the irreducible highest weight representation of highest weight $\gamma$. For integers $m\geq n\geq0$, call a weight $\beta$ of $V(n\rho)$ $\delta$-maximal if it is maximal in its $\delta$-string. The affine $\delta$-maximal component conjecture. If $\lambda=m\rho+\beta$ with $\beta$ a $\delta$-maximal weight of $V(n\rho)$, then

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

Equivalently, the $\delta$-maximal components in this tensor product are precisely those with $\lambda=m\rho+\beta$ for a $\delta$-maximal weight $\beta$ of $V(n\rho)$. The statement is presented as an equivalent affine formulation of the unequal-parameter conjecture, supported by the Virasoro-action calculation; the source does not provide a resolution beyond the partial results stated for affine types.