This question is inspired by Probability of commutation in a compact group, which asked whether $P(xyx^{-1}y^{-1} = 1)$ could take values strictly between $0$ and $1$ on a compact connected group. That question turned out to have a rather easy answer, but one that was quite particular to the word $xyx^{-1}y^{-1}$.

Question:Let $w$ be an arbitrary word in $k$ letters, and let $G$ be a compact connected group. Let $x_1, \dots, x_k$ be drawn uniformly from $G$. Can $P(w(x_1, \dots, x_k)=1)$ be strictly between $0$ and $1$?

By the Peter--Weyl theorem we may assume that $G$ is a compact connected Lie group.

There is some low-hanging fruit, like $w = [[x,y],z]$, but that's still very specialized. Admittedly I am not even clear on the words $w = x^n$.