Caputo, Dai Pra and Posta's dimension-free modified log-Sobolev conjecture for inhomogeneous zero-range processes

Consider an inhomogeneous zero-range process with $n$ sites, rate functions $r_i$, particle number $m$, and site weights $p=(1/n,\ldots,1/n)$. Assume that there are constants $\Delta,\delta>0$ such that, for every $i\in [n]$ and every $\ell\in\mathbb{Z}_+$,

$$\delta\le r_i(\ell+1)-r_i(\ell)\le\Delta.$$

Here $\alpha(Q)$ denotes the modified log-Sobolev constant of the process. Caputo, Dai Pra and Posta's conjecture. Under this assumption, there is a dimension-free constant $c(\delta,\Delta)>0$ such that, for $p=(1/n,\ldots,1/n)$ and any number $m$ of particles,

$$\alpha(Q)\ge c(\delta,\Delta).$$

The conjecture predicts persistence of a dimension-free modified log-Sobolev inequality beyond the perturbative regime $\Delta/\delta<2$, where the known bound $\alpha(Q)\ge 2\delta-\Delta$ becomes trivial. It concerns inhomogeneous rates, for which the cited theorem was presented as the only available criterion in the source.