The k-Schur structure-coefficient bound conjecture

For $k$-bounded partitions $\lambda$, $\mu$, and $\nu$, write the product of $k$-Schur functions as

$$s_{\lambda}^{(k)}[X]s_{\mu}^{(k)}[X]=\sum_\nu {c_{\lambda\mu}^{\nu}}^{(k)}s_\nu^{(k)}[X],$$

where $c_{\lambda\mu}^{\nu}$ are the ordinary Littlewood–Richardson coefficients. The k-Schur structure-coefficient bound conjecture.

$$0\leq {c_{\lambda\mu}^{\nu}}^{(k)}\leq c_{\lambda\mu}^{\nu}.$$

The conjecture asks for nonnegativity together with an upper bound by the classical Littlewood–Richardson coefficient; the text gives integrality and agreement with the classical coefficients for $k\geq|\nu|$, but no general proof.