The Lorentzian conjecture for weight multiplicities of \mathfrak{sl}_m-modules

Let $\Lambda$ be the integral weight lattice of $\mathfrak{sl}_m(\mathbb{C})$, let $\mathrm{V}(\lambda)$ be the irreducible module with highest weight $\lambda$, and let $\mathrm{V}(\lambda)_\mu$ denote its $\mu$-weight space. For distinct $i,j\in[m]$, set $\mu(i,j)=\mu+e_i-e_j$. Weight-multiplicity conjecture. For every $\lambda,\mu\in\Lambda$,

$$(\dim \mathrm{V}(\lambda)_\mu)^2\geq \dim \mathrm{V}(\lambda)_{\mu(i,j)}\dim \mathrm{V}(\lambda)_{\mu(j,i)}.$$

This asserts log-concavity of weight multiplicities along root directions. The source presents it as a conjectural strengthening beyond the established finite-dimensional special cases.