Reflection-symmetry conjecture for LEGO counting polynomials

For each positive integer $n$, let $p_n(w)$ be the polynomial counting the LEGO structures made from $n$ parallel $w\times1$ tiles. Reflection-symmetry conjecture. For every $n\in\mathbb{N}$,

$$p_n(1-w)=(-1)^{n-1}p_n(w).$$

The symmetry would explain the alternating-sum identity for the polynomial coefficients observed in the enumerated data. It is presented as an observation in a conjecture environment and is not proved in the supplied text.