Modular-3 congruence for the conjectural Pascal polynomial factors

Let $P(k)$ be the $k\times k$ Pascal matrix, and let $c_k(t)$ be the polynomial appearing in the prime-power characteristic-polynomial generalization.

Modular-3 conjecture. For every $k$,

$$c_k(t)\equiv (t+1)^{3k}\det(tI+P(k))\pmod 3.$$

Together with the preceding conjecture, this would give recursive formulas for characteristic polynomials of Pascal matrices modulo $3$ and imply that all roots have multiplicative order a power of $2$ in the algebraic closure of $\mathbf{F}_3$. The source gives no resolution status.