Do balanced words have a short period?

We conjecture that each balanced word on N letters - either arises from a balanced word on two letters by expanding both letters with a congruence word, - or is D-periodic with D <= 2^N-1. Our conjecture arises from extensive numerical experiments. It implies, for any fixed N, the finiteness of the number of balanced words on N letters which do not arise from expanding a balanced word on two letters. It refines a theorem of Graham and Hubert, which states that non-periodic balanced words are congruence expansions of balanced words on two letters. It also relates to Fraenkel's conjecture, which states that for N > 2, every balanced word with distinct densities d_1 > d_2 > ... > d_N satisfies d_i = 2^{N-i} / (2^N-1), since this implies that the word is D-periodic with D= 2^N-1. For N < 7, we provide a tentative list of the density vectors of balanced words which do not arise from expanding a balanced word with fewer letters. We prove that the list is complete for N=4 letters.
We also prove that deleting a letter in a congruence word always produces a balanced word and this construction allows us to further reduce the list of density vectors that remains unexplained. Moreover, we prove that deleting a letter in an m-balanced word produces an (m+1)-balanced word, thus extending and simplifying a result of Sano et al. (2004).