King–Tollu–Toumazet stretching conjecture for Littlewood–Richardson coefficients

Let $\lambda$, $\mu$ and $u$ be partitions, and let $c_{\lambda\mu}^{\nu}$ denote the Littlewood–Richardson coefficient. Assume that $c_{\lambda\mu}^{\nu}>0$. For each positive integer $N$, write $N\lambda$, $N\mu$ and $N\nu$ for the partitions obtained by multiplying all parts by $N$. King–Tollu–Toumazet stretching conjecture. There \exists a polynomial $P_{\lambda\mu}^{\nu}(N)$ in $N$ with nonnegative rational coefficients such that

$$P_{\lambda\mu}^{\nu}(0)=1$$

and

$$P_{\lambda\mu}^{\nu}(N)=c_{N\lambda\,N\mu}^{N\nu}$$

for all positive integers $N$. This conjecture predicts a polynomial stretching law for Littlewood–Richardson coefficients; the paper proves that these stretched coefficients are polynomial in $N$ with rational coefficients and gives a degree bound, but does not establish the required nonnegativity of the coefficients.