Polynomial-coefficient conjecture for the difference of special-boundary transfer-matrix eigenvalues

Let $m$ be the lattice-width parameter, let $q$ be the Potts-model state parameter, and let $\lambda_{\star,=}(m)$ and $\lambda_{\star,\neq}(m)$ denote the dominant transfer-matrix eigenvalues for the $=$ and $\neq$ boundary conditions, respectively. For each positive integer $\ell$, let $D_\ell$ be a polynomial in $m$. Polynomial-coefficient conjecture. There exist polynomials $D_1,D_2,\ldots$ with rational coefficients, with $\deg D_\ell=\ell$, such that

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

This conjecture formalizes the empirical observation that the coefficients in the asymptotic expansion of the eigenvalue difference are polynomial in $m$, with the coefficient at order $q^{-\ell}$ having degree exactly $\ell$. The authors report verification for the corresponding computations discussed in the paper; the general assertion remains open.