$\DeclareMathOperator\PGL{PGL}\DeclareMathOperator\Conj{Conj}$Let $G$ be the symmetric group $S_n$ or the projective general linear group $\PGL_2(n)$.
Let $X$ be a cyclically reduced word in the abstract variables $x_1, x_2, \ldots,x_k$, i.e. $X$ is a product containing $x_1, x_2, \ldots,x_k$ and their inverses, without any element appearing next to its own inverse in any cyclic permutation. (Only words with length $4$, $6$, $8$ are needed in my research.)
Consider the probability $P$ that the word sums to $1$, with each $x_i$ chosen uniformly and independently from $G$.
Question:
What are the upper bounds of $\log_{|G|}P$?
If $\log_{|G|}P$ converges when $n\to\infty$, what's the value?
Answers are acceptable for either $G=S_n$ or $G=\PGL_2(n)$.
Known:
If there's a variable occurring only once in $X$, then $P$ is exactly $1/|G|$.
If $X=x_1^k$, then the limit is $-1/k$ for symmetric groups by David E Speyer's argument.
As Richard Stanley pointed out, if $X=x_1x_2x_1^{-1}x_2^{-1}$, then $P=|\Conj(G)|/|G|$. ($|\Conj(G)|$ is the number of conjugacy classes of $G$)
The formula $P=|\Conj(G)|/|G|$ holds for the words $x_1x_1x_2x_2$ and $x_1x_2x_1x_2^{-1}$ if all the characters of $G$ are real, and that's exactly the case for $S_n$ and $\PGL_2(n)$.