A conjecture of M. Billaud is: Given a word w, if, for each letter x occurring in w, the word obtained by erasing all the occurrences of x in w is a fixed point of a nontrivial morphism f_x, then w is also a fixed point of a nontrivial morphism. We prove that this conjecture is equivalent to a similar one on sets of words. Using this equivalence, we solve these conjectures in the particular case where each morphism f_x has only one expansive letter.