The zigzag circulant closed-form conjecture for Speyer's polynomial

For $n\geq 5$, let $C^n_{1,2}$ be the circulant graph obtained from the cycle on $n$ vertices by adding the edges joining each vertex to the vertex two steps away, with indices taken modulo $n$. Let $g_G(t)$ denote Speyer's matroid polynomial. The zigzag circulant conjecture. For every integer $n\geq 5$,

$$g_{C^{n}_{1,2}}(t)=t+\sum_{i=1+\lfloor n/2\rfloor}^{n-2}\frac{n}{i}\binom{i}{n-i}t(1+t)^i+\begin{cases}t(1+t)^2&\text{if $n$ is even,}\\0&\text{if $n$ is odd.}\end{cases}$$

The formula was confirmed computationally for all $n\leq 20$ in the paper and is open beyond those computations.