The two-thirds nonnegativity rule for twisted Verlinde algebras

Let $k$ be an even positive integer, and let $\mathfrak{g}$ be a twisted affine Lie algebra not of type $A_{2\ell}^{(2)}$. Let $P_k^+$ be the set of dominant weights at level $k$, let $\overline{\lambda}$ denote the finite part of a weight, and let $\overset{\circ}{Q}$ be the root lattice of the underlying finite root system. Let $R(\mathfrak{g})$ have basis $\{\chi_{\lambda}\}_{\lambda\in P_k^+}$, with structure constants $N_{\lambda\mu}^{\nu}$. Two-thirds nonnegativity conjecture. If $\lambda,\mu,\nu\in P_k^+$ and at least two of $\overline{\lambda},\overline{\mu},\overline{\nu}$ lie in $\overset{\circ}{Q}$, then

$$N_{\lambda\mu}^{\nu}\geq 0.$$

This observed rule describes a partial nonnegativity pattern in the vector-space-graded algebra $R(\mathfrak{g})$; the supplied text reports experimental evidence but no proof or resolution.