Sparse free energy Lyapunov conjecture for the local-field equation

Let $\nu$ be a probability measure on $\mathbb{R}^3$, let $\bar{\nu}$ be its $(0,1)$-marginal, and let $\pi$ and $\bar{\pi}$ denote the invariant measure of the Markov local-field equation and its corresponding marginal. Define the sparse free energy by

$$\mathbb{H}_2(\nu):=\mathcal{H}(\nu\mid\pi)-\mathcal{H}(\bar{\nu}\mid\bar{\pi}).$$

Sparse free energy Lyapunov conjecture. The sparse free energy $\mathbb{H}_2$ serves as a Lyapunov function for the measure-flow of the local-field equation. The analogous functional is known to be a Lyapunov function for the Markov local-field equation for a large class of potentials, while its Lyapunov property for the local-field equation is proposed as a future direction.