The strong rearrangement bijection conjecture for generalized factor order

Let $\mathbb{P}^*$ be the set of finite words over the positive integers, let $\mathcal{F}(u)$ denote the set of words containing $u$ as a factor in generalized factor order, and let $u\backsim v$ denote Wilf equivalence. A bijection is weight-preserving when it preserves the weight of every word, and a word is a rearrangement of another when it has the same multiset of letters. Strong rearrangement bijection conjecture. If $u\backsim v$, then there is a weight-preserving bijection

$$f:\mathbb{P}^*\longrightarrow\mathbb{P}^*$$

such that for every $w\in\mathbb{P}^*$, $f(w)$ is a rearrangement of $w$ and

$$w\in\mathcal{F}(u)\iff f(w)\in\mathcal{F}(v).$$

This would strengthen the weak rearrangement conjecture by realizing Wilf equivalence through a letter-rearranging bijection, and is left open in the paper.