The weak rearrangement conjecture for generalized factor order

Let $P$ be the positive integers with their usual order, let $\mathbb{P}^*$ denote the set of finite words over $P$, and let $u\backsim v$ mean that the weight generating functions of $u$ and $v$ are equal. Two words are rearrangements when they contain the same letters with the same multiplicities. Weak rearrangement conjecture. If $u\backsim v$, then $v$ is a rearrangement of the letters of $u$. This conjecture is known for words with increasing/decreasing factorizations, but remains open for arbitrary words.