Linear-polynomial coefficient conjecture for Potts-model eigenvalue expansions

Let $q$ be the Potts-model parameter, let $m$ be a positive integer, and let $\lambda_{\star,=}(m)$ and $\lambda_{\star,\neq}(m)$ be the relevant eigenvalues. There exist polynomials $E_1,E_2,\ldots$ and $F_1,F_2,\ldots$ with rational coefficients, each of degree $1$, such that

$$\log\left[(-1)^{m+1}\left(\lambda_{\star,=}(m)-\lambda_{\star,\neq}(m)\right)\right]=\sum_{\ell=1}^{m-1}E_\ell(m)q^{-\ell}+O(q^{-m}),$$ $$-\log\left[(-1)^{m+1}q^m\log\left(\frac{\lambda_{\star,=}(m)}{\lambda_{\star,\neq}(m)}\right)\right]=\sum_{\ell=1}^{m-1}F_\ell(m)q^{-\ell}+O(q^{-m}).$$

Linear-coefficient conjecture. Both displayed expansions hold. The coefficients have been checked computationally in the ranges reported in the paper, and the relations among the associated expansions explain the observed polynomial structure. A general proof remains open.