Bóna–Ju–Ann palindromicity conjecture for circular graph polytopes

Let $G$ have vertex set $V=\{1,2,\ldots,k\}$ and edge set consisting of all pairs $\{i,i+1\}$, with $k+1$ identified with $1$; such a graph is called a circular graph. Let $\ell\in\mathbb{P}$.

Bóna–Ju–Ann conjecture. If $k=2\ell$, then

$$\mathrm{Ehr}(P(G),x)(1-x)^{2\ell+1}$$

is a polynomial in $x$ of degree $2\ell-2$ with symmetric coefficients. If $k=2\ell+1$, then

$$\mathrm{Ehr}(P(G),x)(1-x)^{2\ell+2}(1+x)$$

is a polynomial in $x$ of degree $2\ell$ with symmetric coefficients.

The source states that this conjecture is confirmed using a result from another paper, so it is solved.