Antipalindromic fixed-point classification conjecture for primitive binary morphisms

Let $\varphi$ be a primitive morphism on the binary alphabet $\{0,1\}$, and let ${\boldsymbol{u}}$ be an infinite fixed point of $\varphi$ containing infinitely many antipalindromes. Let $\mathcal{A}_1$ and $\mathcal{A}_2$ be the two classes of morphisms defined in the paper.

Antipalindromic fixed-point classification conjecture. The morphism $\varphi$ or its square $\varphi^2$ is conjugated to a morphism in

$$\mathcal{A}_1\cup\mathcal{A}_2.$$

This is presented as the main open question for morphisms with antipalindromic fixed points. The paper confirms it when $\varphi$ is uniform or when the fixed point is palindromic.