Parker's conjectural coordinator polynomial for cyclotomic lattices of order twice a prime

Let $p$ be an odd prime, let $\zeta_{2p}=e^{2\pi i/(2p)}$, and let $h_{2p}(x)$ be the coordinator polynomial of $\mathbb Z[\zeta_{2p}]$ with respect to the set of all $2p$th roots of unity. Parker's conjecture. The coordinator polynomial is

$$h_{2p}(x)=\sum_{k=0}^{\frac{p-3}{2}}\left(x^k+x^{p-1-k}\right)\sum_{j=0}^{k}\binom pj+x^{\frac{p-1}{2}}\sum_{j=0}^{\frac{p-1}{2}}\binom pj$$ $$=\sum_{k=0}^{\frac{p-3}{2}}\left(x^k+x^{p-1-k}\right)\sum_{j=0}^{k}\binom pj+2^{p-1}x^{\frac{p-1}{2}}.$$

The paper proves this conjecture by computing the relevant cyclotomic polytope data.