Extension of the co-finite locally compact case to separated infinite complements

Determine whether a regular space E remains a C-space under the hypothesis that E contains a locally compact subspace G whose complement H = E \ G is separated and infinite (i.e., H admits pairwise disjoint open neighborhoods separating its points).

Background

Proposition 19 establishes that if a regular space E has a co-finite locally compact subspace G (so the complement H is finite), then E is a C-space. The authors investigate broader scenarios where E has a large locally compact subspace but with an infinite remainder.

They state uncertainty about whether this conclusion extends when H = E \ G is separated and infinite, which would considerably generalize the co-finite case and tie into the broader program of characterizing C-spaces beyond local compactness.