Monomial-factor conjecture for the stationary distribution

Let $S_n$ be the symmetric group and let $\zeta(w)$ be the normalized stationary-distribution component defined by the nullspace of $P^T-I$. Assume that each $\zeta(w)$ is a polynomial, and let $\eta(w)$ denote the largest monomial that can be factored out of $\zeta(w)$. Let $[m]=\{0,1,2,\ldots,m\}$. Monomial-factor conjecture. The map $w\mapsto\eta(w)$ is an $n$-to-$1$ map from $S_n$ to

$$\left\{x_1^{a_1+a_2+\cdots+a_{n-2}}x_2^{a_2+\cdots+a_{n-2}}\cdots x_{n-2}^{a_{n-2}}\mathrel{|}(a_1,a_2,\ldots,a_{n-2})\in[n-2]\times[n-3]\times\cdots\times[1]\right\}.$$

Moreover, if

$$\eta(w)=x_1^{a_1+a_2+\cdots+a_{n-2}}x_2^{a_2+\cdots+a_{n-2}}\cdots x_{n-2}^{a_{n-2}},$$

then

$$a_i=\#\{k\in[i+2,n]\mid w_k\in[w_i,w_{i+1}]\},$$

where $[w_i,w_{i+1}]$ denotes a cyclic subinterval of $[n]$.

This gives a proposed explicit classification of the monomial factors of the stationary-distribution components; the source does not state whether it has been proved or disproved.