Random length-spectrum rigidity for free groups

We say that a subset $S\subseteq F_N$ is spectrally rigid if whenever $T_1, T_2\in \mathrm{cv}_N$ are points of the (unprojectivized) outer space such that $\vert\vert g\vert\vert _{T_1}=\vert\vert g\vert\vert _{T_2}$ for every $g\in S$, then $T_1=T_2$ in $\mathrm{cv}_N$. It is well known that $F_N$ itself is spectrally rigid; it also follows from the result of Smillie and Vogtmann that there does not exist a finite spectrally rigid subset of $F_N$. We prove that if $A$ is a free basis of $F_N$ (where $N\ge 2$), then almost every trajectory of a non-backtracking simple random walk on $F_N$ with respect to $A$ is a spectrally rigid subset of $F_N$.