Hof–Knill–Simon conjecture on palindromic fixed points and conjugates in class P

Let ${\bf u}$ be the fixed point of a primitive morphism. An infinite sequence is palindromic if it contains an infinite number of palindrome factors.

Hof–Knill–Simon conjecture. ${\bf u}$ is palindromic if and only if there exists a morphism $\varphi\neq\operatorname{Id}$ such that

$$\varphi({\bf u})={\bf u}$$

and $\varphi$ has a conjugate in class $\mathcal P$.

This conjecture was reformulated from the original question of Hof, Knill and Simon about whether every palindromic fixed point arises from a substitution in an extended version of class $\mathcal P$. The paper gives a counterexample on a ternary alphabet, so the conjecture is refuted.