Bipartite maximizer and clique free-energy conjectures for the antiferromagnetic Ising model

Let $G$ range over $\Delta$-regular graphs, let $\alpha_G(B,\lambda)$ be the occupancy fraction, and let $F_G(B,\lambda)$ be the free energy density. Bipartite maximizer and clique free-energy conjectures. For all $\Delta\geq 3$ and all $(B,\lambda)\in[0,1]^2$:

1. The complete bipartite graph $K_{\Delta,\Delta}$ maximizes $\alpha_G(B,\lambda)$ over $\Delta$-regular graphs.
2. The complete graph $K_{\Delta+1}$ minimizes $F_G(B,\lambda)$ over $\Delta$-regular graphs.

The first assertion extends the proved $3$-regular result for $K_{3,3}$, while the second concerns free-energy minimization despite the failure of clique occupancy minimization throughout the full parameter space. The supplied text does not state whether either assertion has been resolved.