If $G=(V,E)$ is a simple, undirected graph, then we define its Hadwiger graph $\textrm{Hadw}(G)$ in the following way:
- $V(\textrm{Hadw}(G)) = \{S\subseteq (V(G): S\neq \emptyset\textrm{ and } S \textrm{ is connected}\}$;
- $E(\textrm{Hadw}(G)) = \{\{S,T\}\subseteq V(\textrm{Hadw}(G)): S\cap T = \emptyset \textrm{ and } (\exists s\in S, t\in T): \{s,t\}\in E(G) \}.$
$G$ embeds to $\textrm{Hadw}(G)$ by $v\mapsto \{v\}$, and the Hadwiger conjecture states $\chi(G) \leq \omega(\textrm{Hadw}(G))$.
Question: If $G, H$ are infinite connected graphs, does $\textrm{Hadw}(G) \cong \textrm{Hadw}(H)$ imply $G\cong H$?
(This is a follow-up to Isomorphic Hadwiger graphs.)
