The one percent property for free groups

We prove that for every $\epsilon >0$, there is $\delta >0$ such that for every probability measure $\mu$ on a free group $\mathbb{F}$, if the operator norm of the action of $\mu$ on $L^2\mathbb{F}$ by convolution exceeds $\epsilon$, then there is a coset of an abelian subgroup of $\mathbb{F}$ with $\mu$-mass at least $\delta$.