Determine whether finite-specific-free-energy ergodic measures are minimal

Determine whether every ergodic gradient Gibbs measure with finite specific free energy and slope in the interior of the admissible slope set is a minimal gradient Gibbs measure.

Background

Minimality means attaining the surface tension, or equivalently the least specific free energy among translation-invariant gradient measures with a fixed slope. The question asks whether finite free energy together with ergodicity is sufficient to force this variational minimality.

The paper identifies this as another conjecture from the general theory. It remarks that the assertion is proved for Lipschitz height functions and isotropic interactions, but leaves it unresolved for the general class of convex interactions considered here.

References

Additionally,Conjecture 10.3.1 conjectures that if $\mu\nabla$ is ergodic, $SFE(\mu\nabla)<\infty$ and $\mathfrak{s}=slope(\mu\nabla)\in\mathfrak{S}\circ$ then $\mu\nabla$ is minimal.

— On the rigidity of sloped height functions in $d\ge 3$ and non-crossing surfaces  (2609.01499 - Adhikari et al., 1 Sep 2026) in Section 2, Subsection 2.4, Infinite free energy and non-minimality