Gruenhage's three-element basis conjecture for uncountable regular Hausdorff spaces

A basis of topological spaces means that every space in the class has one of the listed spaces as the relevant basis representative. Gruenhage's three-element basis conjecture. The uncountable regular Hausdorff spaces have a three-element basis consisting of a set of reals of cardinality $\aleph_1$ with the metric topology, the Sorgenfrey topology, and the discrete topology.

This is a proposed classification of uncountable regular Hausdorff spaces by three topological models; the supplied material gives no resolution.