The Pascal descent polynomial twin and anti-twin conjecture

Let $[n]=\{1,\ldots,n\}$, and let $I,J\subseteq[n]$ have the same cardinality $s\leq\lfloor n/2\rfloor$. Let $p_n(I)$ and $p_n(J)$ be the Pascal descent polynomials associated with these subsets, and let $\tau_n$ send each position $i$ to $n+1-i$. Assume $p_n(I) eq0$ and $p_n(J) eq0$.

Pascal descent polynomial conjecture. One has

$$p_n(I)=p_n(J)$$

if and only if $I=J$, or $n$ is odd and $I=\tau_n(J)$; and

$$p_n(I)=-p_n(J)$$

if and only if $n$ is even and $I=\tau_n(J)$.

This is the Pascal-descent-polynomial reduction of the tabloid conjecture and would classify the relevant polynomial coincidences up to reversal and sign. The source gives no resolution status.