The circulant graph conjecture for Speyer's polynomial

For integers $n\geq 1$ and $k$, let $C^n_{1,k}$ be the graph on vertices identified with $\{1,\ldots,n\}$, obtained from the cycle by adding the edges $\{i,i+k\}$, with vertex labels taken modulo $n$. Let $\mathsf{N}_2(G)$ denote the coefficient of $t^2$ in Speyer's matroid polynomial $g_G(t)$. The circulant graph conjecture. For every integer $n\geq 5$,

$$\mathsf{N}_2\big(C^{2n}_{1,n-1}\big)=2^{n-1}-n.$$

The formula is supported by computations for the cases tested in the paper, but remains conjectural in general.