Polynomial-ratio conjecture for repeated-entry generalized monotone triangles

Let $\alpha$ be the generalized monotone-triangle enumeration, let $A_{n-1}:=\alpha(n-1;1,2,\ldots,n-1)$, and let $n\geq k\geq4$. Polynomial-ratio conjecture. There exist polynomials $p_k(n)$ and $q_k(n)$ satisfying

$$\deg p_k-\deg q_k=k-3$$

and

$$\alpha(n+1;1,2,\ldots,k-3,k-1,k,k-1,k,\ldots,n)=\frac{p_k(n)}{q_k(n)}A_{n-1}.$$

The source gives two lower-index computational examples before formulating this general conjecture; it supplies no proof or resolution.