  The Rearrangement Conjecture states that if two words over $\mathbb{P}$ are
Wilf-equivalent in the factor order on $\mathbb{P}^\ast$ then they are
rearrangements of each other. We introduce the notion of strong
Wilf-equivalence and prove that if two words over $\mathbb{P}$ are strongly
Wilf-equivalent then they are rearrangements of each other. We further
conjecture that Wilf-equivalence implies strong Wilf-equivalence.
