Simultaneous asymptotic equidistribution conjecture for free word tuples

Let $\mathbb F_r$ be the free group on $r$ generators. A tuple $(w_1,\ldots,w_d)\in\mathbb F_r^d$ is simultaneously asymptotically equidistributed when its word-value distribution on $S_n^d$ approaches the equidistribution as $n\to\infty$. Simultaneous equidistribution conjecture. Let $d\in\mathbb N$ and $w_1,\ldots,w_d\in\mathbb F_r$. If $\langle w_1,\ldots,w_d\rangle$ generates a free subgroup of rank $d$ and this subgroup is not contained in a subgroup of smaller rank, then $(w_1,\ldots,w_d)$ is simultaneously asymptotically equidistributed. The source notes that rank obstructions are necessary and presents this as the apparent remaining condition, but does not provide a proof.