This question is related to that (if $s$ is co-prime with prime $p$ and a permutation in $S_s$ has order $p$, then it fixes a point).

Let us fix two (finite) numbers $p\gg 1, n\gg 1$. Say, $p=47, n=18999$. Take a sequence $s_1,s_2,...$ of numbers not divisible by $p$. For each $s$ compute $p(s)$ as follows. Take two permutations $a, b$ in the symmetric group $S_s$ such that $\langle a,b\rangle$ is highly (say, $3$-)transitive on $\{1,...,s\}$. Compute the probability $p(s,a,b)$ that for a word $w(x,y)$ of length $n$, $w(a,b)$ has a fixed point. Then $p(s)$ is the maximum of $p(s,a,b)$ for all (such) $a,b$.

**Question.** Is it true that $$\lim_{i\to\infty} p(s_i) = 0?$$

** Note. ** As Doron Puder explained to me, one cannot replace "highly transitive" by "transitive" because $a,b$ can generate a dihedral group of order $2s_i$ in which case $p(s_i,a,b)$ will be at least $1/5$. F. Ladisch explains in his answer to the previous version of the question below that "2-transitive" is not enough either.

** Update** Perhaps the "correct" condition instead of "highly transitive" should be "$\langle a,b\rangle$ contains "few" permutations having fixed points, that is the portion of such permutations in $\langle a,b\rangle$ tends to 0 as $i\to \infty$. Examples of Puder and Ladisch contain "many" permutations fixing a point and the arguments were based on that fact. Thus the problem is this: if for many words $w(x,y)$ of length $n$ the permutation $w(a,b)$ has a fixed point, then many elements in $\langle a,b\rangle$ have fixed points, a local-to-global property.