What is the length of the shortest word $w\in F_2$ such that $w(x,y)$ is trivial for every $x,y\in S_n$?

There is a simple argument showing that we must have $\ell(w)\geq n$. See here for instance. It seems likely however that $\ell(w)$ must be super-polynomial in $n$.

What is the state of the art for lower bounds?

(As for upper bounds, the paper "Identical relations in symmetric groups and separating words with reversible automata" by Gimadeev and Vyalyi finds an upper bound of $\exp(\sqrt{n}\log n)$.)