Bruhn's Hamilton-circle conjecture for locally finite planar graphs

Let $G$ be a locally finite planar graph. A Hamilton circle is a circle in the topological space $|G|$ that visits every vertex of $G$.

Bruhn's conjecture. Every $4$-connected locally finite planar graph has a Hamilton circle.

This is the infinite analogue of Tutte's theorem that every finite $4$-connected planar graph has a Hamilton cycle. The supplied status evidence concerns a different Hamilton-circle theorem, so no direct resolution of this conjecture is established here.