  In [A. Frid, S. Puzynina, L.Q. Zamboni, \textit{On palindromic factorization
of words}, Adv. in Appl. Math. 50 (2013), 737-748], it was conjectured that any
infinite word whose palindromic lengths of factors are bounded is ultimately
periodic. We introduce variants of this conjecture and prove this conjecture in
particular cases. Especially we introduce left and right greedy palindromic
lengths. These lengths are always greater than or equals to the initial
palindromic length. When the greedy left (or right) palindromic lengths of
prefixes of a word are bounded then this word is ultimately periodic.
