Let $n \in \mathbb N$ and let $\sigma,\tau \in {\rm Sym}(n)$. I am looking for a permutation $x \in {\rm Sym}(n)$ that minimizes the Hamming distance between $x^2 \sigma$ and $\tau x$. Here, the Hamming distance between permutations $a,b \in {\rm Sym}(n)$ is defined as:
$$d(a,b) = \#\{1 \leq i \leq n \mid a(i) \neq b(i) \}.$$
Conjecture: For all $\varepsilon>0$, there exists $n_0 \in \mathbb N$, such that for all $n \geq n_0$ and $\sigma,\tau \in {\rm Sym}(n)$, there exists $x \in {\rm Sym}(n)$ with $$d(x^2 \sigma,\tau x)\leq \varepsilon n.$$
I have tried to find solutions to this problem for $n=50$ and random $\sigma,\tau$ with various approaches. The outcome was that $x \in {\rm Sym}(n)$ with $d(x^2 \sigma,\tau x)\leq 8$ could be found in almost all cases. I found this encouraging but of course it does not mean much with regard to the conjecture - but maybe it indicates that this is an interesting problem.
Of course there are many possible generalizations of this question.
