Free-energy asymptotic expansion conjecture for the square-lattice Potts eigenvalues

Let $q$ be the Potts-model parameter, let $m$ be a positive integer, and let $\lambda_{\star,\neq}(m)$ and $\lambda_{\star,=}(m)$ denote the relevant dominant eigenvalues for the $\neq$ and $=$ boundary conditions. There exist polynomials $C_1,C_2,\ldots$ with rational coefficients, each of degree $1$, such that

$$\log \frac{\lambda_{\star,\neq}(m)}{q^m}=\sum_{k=1}^m C_k(m)q^{-k}+O(q^{-(m+1)}),$$ $$\log \frac{\lambda_{\star,=}(m)}{q^m}=\sum_{k=1}^m C_k(m)q^{-k}+(-1)^{m+1}q^{-m}+O(q^{-(m+1)}).$$

Free-energy expansion conjecture. The two displayed asymptotic expansions hold. This has been verified for $m\le 10$ using the explicitly computed polynomials $C_1,\ldots,C_{10}$. The conjecture refines the proved leading-order statement for the difference of the two eigenvalues, but remains unproved in general.