Halin's end degree conjecture

Let $G$ be a graph and let an end of $G$ be an equivalence class of rays, where two rays are equivalent if there are infinitely many vertex-disjoint paths between them. The degree of an end is the maximum cardinality of a collection of pairwise disjoint rays in that equivalence class. Given a set of disjoint equivalent rays in $G$, a ray graph is a graph whose vertex set is that set of rays, with an edge $RS$ whenever there are infinitely many disjoint independent $R$--$S$ paths in $G$. A ray graph for an end is a connected ray graph on a degree-witnessing subset of that end.

Halin's conjecture. Every graph contains ray graphs for all its ends.

For ends of finite degree, the assertion is straightforward, and it is also known for ends of countably infinite degree, where the ray graph can be chosen as a ray using Halin's grid theorem. The conjecture remains open for ends of uncountable degree.