Fulton's multiplicity-one conjecture for tensor products of GL(n) representations

Let $\lambda,\mu,\nu \in ({\mathbb Z}^n)^3$ be a triple of dominant weights for $\operatorname{GL}_n({\mathbb C})$, and let $V_\lambda,V_\mu,V_\nu$ be the corresponding irreducible representations. Fulton's conjecture. If $V_\nu$ occurs exactly once as a constituent of $V_\lambda\otimes V_\mu$, then for every $N\in{\mathbb N}$, $V_{N\nu}$ occurs exactly once as a constituent of $V_{N\lambda}\otimes V_{N\mu}$. This multiplicity-one assertion concerns the stability of Littlewood–Richardson coefficients under simultaneous scaling of the highest weights. The source presents it as a conjecture proposed by William Fulton; its resolution is indicated by the source's comparison with the proven saturation conjecture.