Converses to generalized Conway--Gordon type congruences

It is known that for every spatial complete graph on $n\ge 7$ vertices, the summation of the second coefficients of the Conway polynomials over the Hamiltonian knots is congruent to $r_n$ modulo $(n-5)!$, where $r_n=(n-5)!\mathrm{/}2$ if $n=8k,8k+7$, and $0$ if $n\mathrm{\ne }8k,8k+7$. In particular the case of $n=7$ is famous as the Conway–Gordon $K_7$ theorem. In this paper, conversely, we show that every integer $(n-5)!q+r_n$ is realized as the summation of the second coefficients of the Conway polynomials over the Hamiltonian knots in some spatial complete graph on $n$ vertices.