Characteristic-polynomial congruences at prime powers congruent to 2 modulo 3

Let $q=p^l$ be a prime power with $q\equiv2\pmod 3$, and let $\chi_n(t)$ denote the characteristic polynomial associated with the Pascal matrix $P(n)$.

Prime-power congruence conjecture. One has

$$\chi_{(q+1)/3}(t)\equiv (t+1)^{(q+1)/3}\pmod p$$

and

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

These congruences predict explicit factorizations in the exceptional residue class $q\equiv2\pmod3$; the source gives no resolution status.