Grytczuk et al.'s finiteness conjecture for extremal pattern-avoiding words

Let $P$ be an avoidable pattern, meaning that some alphabet admits arbitrarily long words avoiding $P$. For a fixed alphabet, an extremal $P$-avoiding word is a word that avoids $P$, but every extension obtained by inserting one letter contains $P$. Grytczuk et al.'s finiteness conjecture. For every avoidable pattern $P$, there exists a constant $k(P)$ such that the set of extremal $P$-avoiding words over a $k(P)$-letter alphabet is finite.

This conjecture proposes a finite extremal analogue of the theory of avoidable patterns. The paper uses it as motivation for studying extremal words for several other pattern classes; its resolution is not given here.