The Rearrangement Conjecture for generalized factor order

Let $P=\mathbb{P}$ be the positive integers with their usual order, and let $u,v\in\mathbb{P}^*$ be words. Two words are Wilf equivalent if they are avoided by the same number of words of every length and sum; equivalently, if

$$F_u(x,y)=F_v(x,y),$$

where

$$F_u(x,y)=\sum_{\substack{w:\,u\text{ is not a generalized factor of }w}}x^{|w|}y^{\|w\|}.$$

Rearrangement Conjecture. If $u$ and $v$ are Wilf equivalent, then they are rearrangements of each other; that is, they have the same multiset of letters. The converse is false: for example, $A_{132}(x,y,0)\ne A_{312}(x,y,0)$. The conjecture asks whether Wilf equivalence forces equality of the letter multisets; it was disproved in this paper, while a weakening for strongly Wilf equivalent words is known.