The complete bipartite and 4-partite Speyer polynomial formulas

For $n\geq3$, let $K_{3,n}=\overline{K}_3\vee\overline{K}_n$ and $K_{1,1,1,n}\cong K_3\vee\overline{K}_n$.

Complete multipartite formula conjecture. Their Speyer polynomials are

$$g_{K_{3,n}}(t)=t(nt^2+nt+t+2)(2+t)^{n-1}-3t(1+t)^{n+1}$$

and

$$g_{K_{1,1,1,n}}(t)=t(nt^2+nt+t+2)(2+t)^{n-1}.$$

These formulas imply the quadratic growth $\mathsf{N}_2(K_{3,n})=\mathsf{N}_2(K_{1,1,1,n})=-n(n-3)/2$ and are supported by the computations in the paper.