The prism and Möbius ladder formulas for Speyer's polynomial

For $n\geq2$, let $K_2\times C_n$ be the prism graph and let $C^{2n}_{1,n}$ be the Möbius ladder, both on $2n$ vertices.

Prism and Möbius ladder conjecture. Their Speyer polynomials are

$$g_{K_2\times C_n}(t)=t\left(1+(1+t)^2+(2^n-n-3)(1+t)^n\right)$$

and

$$g_{C^{2n}_{1,n}}(t)=t\left(1+(2^n-n-1)(1+t)^n\right).$$

The formulas were confirmed computationally for $n\leq10$ and imply the stated values of $\mathsf{N}_2$ for these families.