Avoiding conjugacy classes on the 5-letter alphabet

We construct an infinite word $w$ over the $5$-letter alphabet such that for every factor $f$ of $w$ of length at least two, there exists a cyclic permutation of $f$ that is not a factor of $w$. In other words, $w$ does not contain a non-trivial conjugacy class. This proves the conjecture in Gamard et al. [TCS 2018]