The coefficient-shape conjecture for $P_k(t,x)$

Write

$$P_k(t,x)=x\left(p_{k,0}(t)x^{k-1}-p_{k,1}(t)x^{k-2}+\cdots+(-1)^{k-1}p_{k,k-1}(t)\right).$$

Thus $p_{k,j}(t)$ denotes the coefficient functions in this expansion. Coefficient-shape conjecture. The coefficients satisfy

$$p_{k,j}(t)=\frac{r_{k,j}(t)}{t^j},$$

where

$$r_{k,j}(t)=\sum_{q=0}^{m}\alpha_{k,j,q}t^{2q}(t+1)^{j-3q},$$

with rational positive nonzero numbers $\alpha_{k,j,q}$; here $m\geq 0$ and $j=3m+l$ for some remainder $l$ on division of $j$ by $3$. The claim is based on calculations of the displayed polynomials $P_k(t,x)$; no proof or resolution is given in the source.