Fish–Reyzin's coupled-walk meeting conjecture for random DFAs

Let $(V,\mathcal C,\{f_x\}_{x\in V})$ be a random DFA with $|V|=n$. For a fixed $t\in\mathbb N$, let $\mathbf Q=\mathbf Q_t$ be the uniform distribution over words in $\mathcal C^t$, and let $W_t$ be a random word sampled according to $\mathbf Q$. For states $x,y\in V$, write $W_t(x)$ and $W_t(y)$ for the states reached by applying the same word to $x$ and $y$. Fish–Reyzin's conjecture. There exists a constant $c>0$ such that, for any pair $(x,y)=(x_n,y_n)\in V^2$ and every $b>0$, with high probability,

$$\mathbf Q\left(W_{cn}(x)\ne W_{cn}(y)\right)\le n^{-b}.$$

This is an open problem arising from average-case reconstruction of random acceptors: applying one common random word is expected to synchronize any pair of states after a linear number of letters, with probability smaller than every polynomial inverse in $n$.