A conjecture attributed to M. Billaud states that if a word is morphically primitive, then at least one of its "heirs" δ_x(w) is also morphically primitive. Here δ_x denotes the morphism deleting the letter x (and being identity otherwise).

Perhaps more natural is the original formulation: if δ_x(w) is morphically imprimitive for all letters x occurring in w, then also w is morphically imprimitive.

The conjecture was formulated in 1993 and is believed to be difficult. It easily holds if the alphabet of w has cardinality at most three (Zimmermann, 1993).

Some special cases were solved in
F. Levé and G. Richomme. On a conjecture about finite fixed points of morphisms, Theor. Comp. Sci. 339 (2005) 103-128

Some insight into morphical (im)primitivity, so far insufficient to solve the conjecture, was recently obtained in
Daniel Reidenbach and Johannes C. Schneider. Morphically primitive words, Theor. Comp. Sci. 410 (2009) 2148-2161
and in
Štěpán Holub, Polynomial algorithm for fixed points of nontrivial morphisms, Discrete Mathematics 309 (2009), 5069-5076

-- StepanHolub - 02 Sep 2011