King–Tollu–Toumazet positivity conjecture for stretched Littlewood–Richardson coefficients

Let $\lambda,\mu,\nu$ be partitions with $|\lambda|+|\mu|=|\nu|$, and let $c^{\nu}_{\lambda\mu}$ be the Littlewood–Richardson coefficient, the multiplicity of the Schur function $s_\nu$ in $s_\lambda s_\mu$. For a positive integer $t$, write $t\lambda=(t\lambda_1,t\lambda_2,\dots)$. Define the stretched coefficient polynomial by

$$P^{\nu}_{\lambda\mu}(t):=c^{t\nu}_{t\lambda,t\mu}.$$

By a theorem of Derksen and Weyman, this agrees for all $t\geq 1$ with a polynomial in $t$. King–Tollu–Toumazet positivity conjecture. Every coefficient of $P^{\nu}_{\lambda\mu}(t)$, in the monomial basis $1,t,t^2,\dots$, is nonnegative. This is a strong positivity conjecture for stretched Littlewood–Richardson coefficients; beyond the cases established in the paper, it remains open and is listed as an unsolved problem in the FrontierMath collection.