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 ≥ 3, 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.