Non-degeneracy of conditional limits for multi-spin systems with uniform mark distribution

Let $\mathcal{X}$ be a finite set, let $h$ be an edge potential, and let $\nu=\mathsf{Unif}(\mathcal{X})$. For $c\in (\kappa h_{\min},\kappa h_{\max})$, let $U_n$ denote the relevant Gibbs measure conditioned on $\operatorname{Conset}_n(h,c)$, and let $\operatorname{Gibbsnondeg}(c)$ be the set of non-degenerate Gibbs measures defined by the preceding construction.

Non-degeneracy conjecture. For every $c\in (\kappa h_{\min},\kappa h_{\max})$, $U_n$ given $\operatorname{Conset}_n(h,c)$ lies asymptotically in $\operatorname{Gibbsnondeg}(c)$.

The conjecture proposes that conditional limits are always non-degenerate for multi-spin systems with a uniform mark distribution, extending the proved result for two-spin systems. The source presents this as an open question; the general case remains unresolved.