Class P' conjecture for primitive morphism fixed points

Let ${\mathbf u}$ be a palindromic fixed point of a primitive morphism $\varphi$. A morphism is said to be of class $P'$ if it has the class-$P'$ property used in the source. Its fixed point is required to have the same language as ${\mathbf u}$.

Class $P'$ conjecture. There exists a morphism of class $P'$ such that its fixed point has the same language as ${\mathbf u}$.

The conjecture is partially resolved: the binary case is solved, and affirmative answers are known for several special classes of morphisms, including marked morphisms and codings of a non-degenerate exchange of three intervals. The general case over larger alphabets remains open.