Classification conjecture for primitive morphism fixed points among binary generalized pseudostandard words

Let a binary generalized pseudostandard word be an infinite word over a binary alphabet generated by generalized pseudopalindromic closure, and call it aperiodic if it is not ultimately periodic. A word is a fixed point of a primitive morphism if it is unchanged by a morphism whose incidence matrix has a positive power with all entries positive. Let $\varphi_k$ denote the morphism defined in the source.

Classification conjecture. Let $\mathbf u$ be an aperiodic binary generalized pseudostandard word, not standard Sturmian, being a fixed point of a primitive morphism. Then

$$\mathbf u=\varphi_k(\mathbf u)$$

for some $k\in\mathbb N$.

The conjecture asserts that the morphisms $\varphi_k$ provide the only class of primitive morphisms with such fixed points, apart from standard Sturmian words. The source presents this as an open problem based on computer experiments.