Generalization of the prime-power characteristic-polynomial formula

Let $\chi_n(t)$ denote the characteristic polynomial associated with the Pascal matrix $P(n)$. For each integer $k\geq 0$, consider a monic polynomial $c_k(t)\in\mathbf{Z}[t]$ of degree $4k$ satisfying

$$c_k(t)=t^{4k}c_k(t^{-1}).$$

If $q$ is a power of a prime $p$ and $0\leq k\leq q/2$, let $\epsilon(q)\in\{-1,0,1\}$ satisfy $\epsilon(q)\equiv q\pmod 3$.

The prime-power generalization conjecture. For every integer $k\geq0$, there exists such a polynomial $c_k(t)$ for which

$$\chi_{q+k}(t)\equiv(t^2+t+1)^{(q-\epsilon(q))/3-k}(t-1)^{(q+2\epsilon(q))/3-k}c_k(t)\pmod p.$$

This conjecture extends the paper's characteristic-polynomial formula for prime-power matrix sizes and predicts a reciprocal polynomial factor independent of $q$; its status is not specified in the source.