The polynomial-denominator conjecture for generalized factor order

Let $\mathbb{P}^*$ be the set of finite words over the positive integers, let $S(u;t,x)$ be the associated generating function, and let an increasing/decreasing factorization mean the factorization used in the paper. Polynomial-denominator conjecture. For every $u\in\mathbb{P}^*$,

$$S(u;t,x)=\frac{x^s t^r}{P(u;t,x)},$$

where $s$ and $r$ are integers and $P(u;t,x)$ is a polynomial, if and only if $u$ has an increasing/decreasing factorization. The claim is motivated by the limitations of the preceding technique and by experimental evidence; its general validity remains open.