Hamilton-circle conjecture from 2-connected radius-two balls

Let $G$ be a connected, infinite, locally finite graph. A 2-connected ball of radius $2$ is the subgraph induced by the vertices in such a ball that is 2-connected. For a path $uwv$ with $uv\notin E(G)$, let $d_G(u)$ and $d_G(v)$ denote the degrees of its endpoints, and let $M_2(w)$ denote the set of vertices at distance at most $2$ from $w$.

Hamilton-circle conjecture. The graph $G$ has a Hamilton circle if every ball of radius $2$ in $G$ is 2-connected and

$$d_G(u)+d_G(v)\geq |M_2(w)|-1$$

for each path $uwv$ with $uv\notin E(G)$.

This conjecture combines local 2-connectivity with a degree condition modeled on the finite Hamiltonicity criterion. The source explicitly says that the authors believe it is true but gives no proof or resolution, so it remains open.