Ferromagnetic comparison conjecture for the log-Sobolev constant

Let $H$ be a Hamiltonian with associated Gibbs measure $\mu_{\Lambda}$ satisfying the conditions in the setting, and let $H_{\text{fer}}$ be a Hamiltonian with associated Gibbs measure $\mu_{\Lambda,\text{fer}}$. Assume that the single-site potentials of $H$ and $H_{\text{fer}}$ agree, while the interaction terms of $H_{\text{fer}}$ are $-|M_{ij}|$ for $i\neq j$. Suppose that $\mu_{\Lambda,\text{fer}}$ satisfies a log-Sobolev inequality with constant $\varrho_{\text{fer}}$.

Ferromagnetic comparison conjecture. The measure $\mu_{\Lambda}$ satisfies a log-Sobolev inequality with constant $\varrho$ such that

$$\varrho\geq\varrho_{\text{fer}}.$$

This conjecture asserts that the log-Sobolev constant of a non-ferromagnetic system is bounded below by that of the associated ferromagnetic system. The paper notes that the conjecture is only partially answered by its main theorem; its general status is open.