Coskey–Farah conjecture on trivial homeomorphisms of Čech–Stone remainders

Let $X$ and $Y$ be second countable, locally compact, noncompact spaces. Write $X^*=\beta X\setminus X$ and $Y^*=\beta Y\setminus Y$, where $\beta X$ and $\beta Y$ are their Čech–Stone compactifications. A homeomorphism between these remainders is trivial when it is induced outside compact subsets by a homeomorphism between the underlying spaces. Coskey–Farah conjecture. $\mathrm{CH}$ implies that, whenever $X^*$ and $Y^*$ are homeomorphic, there are nontrivial homeomorphisms between them, while $\mathrm{PFA}$ implies that all homeomorphisms between $X^*$ and $Y^*$ are trivial. This conjecture extends the corresponding question about automorphisms of $\mathcal P(\mathbb N)/\operatorname{Fin}$ and connects set-theoretic axioms with the structure of abelian corona algebras; its resolution is not supplied here.