In a 2013 paper with S. Puzynina and L. Zamboni, we stated the following conjecture: Let $w$ be an aperiodic infinite word, that is, a sequence of symbols not ending by an infinite repetition of the form $uuu\cdots$. Then for every $K>0$ there exists a prefix of $w$ which cannot be decomposed into a concatenation of less than $K$ different palindromes.

The conjecture stays unproven, but I will give a survey on recent advances on it and possible methods for the general proof.