PRV for the Fusion product, the case $\lambda\gg\mu$

Given a complex simple Lie algebra $\mathfrak{g}$ and a positive integer $\ell$, under the assumption $\lambda\gg\mu$, we show that irreducible representations of $\mathfrak{g}$ of the form $V(\lambda +w\mu)$, $w\in W,$ with level at most $\ell$ appear in the fusion product of $V(\lambda)$ and $V(\mu)$. This verifies the existence of PRV components for the fusion product in this special case.