Reiner–Tenner–Yong's Fomin–Kirilov polynomial ratio conjecture

Let $w$ be a dominant permutation, meaning a 132-avoiding permutation, whose Lehmer code $\lambda=(c_1(w),\ldots,c_n(w))$ is the rectangular staircase shape $\lambda=\delta_d(b^a)$, where $\delta_d=(d-1,d-2,\ldots,1)$ and $\delta_d(b^a)$ is obtained by replacing each square by an $a\times b$ rectangle. Let $\ell(w)$ denote the length of $w$, and let $FK(w,\ell)$ be the polynomial defined by summing $(x+i_1)\cdots(x+i_\ell)$ over all 0-Hecke words $(s_{i_1},\ldots,s_{i_\ell})$ of $w$ of length $\ell$. Reiner, Tenner and Yong's equivalent conjecture.

$$\frac{FK(w,\ell(w)+1)}{FK(w,\ell(w))}=\binom{\ell(w)+1}{2}\left(\frac{4x}{d(a+b)}+1\right).$$

The source states that this assertion is equivalent to the barely set-valued tableau enumeration conjecture above. Since that conjecture is proved in the paper, this equivalent formulation is solved as well.