Halin's ray graph conjecture

Let $G$ be a graph and let $\theta$ be an end of $G$. A family of paths is independent if its paths intersect only at their endpoints. For a set $\mathcal{R}$ of pairwise disjoint equivalent rays in $G$ and a family $\mathcal{P}$ of independent paths whose endpoints lie precisely on the rays of $\mathcal{R}$, define the ray graph associated with $\mathcal{R}$ and $\mathcal{P}$ to have one vertex for each ray in $\mathcal{R}$, with two vertices adjacent exactly when infinitely many paths in $\mathcal{P}$ connect the corresponding rays. A ray graph for an end is a connected ray graph constructed on a subset of that end witnessing its degree.

Halin's ray graph conjecture. Every graph admits a ray graph for each of its ends.

This conjecture seeks to extend the structural theory of ends beyond Halin's grid theorem by ensuring that every end has a connected ray graph witnessing its degree. Its status is not resolved in the supplied source.