The PFA triviality conjecture for Stone–Čech remainders

Let $X$ and $Y$ be locally compact Polish spaces. For a locally compact space $X$, write $X^* = \beta X\setminus X$ for its Stone–Čech remainder, and call a homeomorphism between such remainders trivial when it is induced by the representation described earlier in the paper. PFA triviality conjecture. Assuming the Proper Forcing Axiom, every homeomorphism between $X^*$ and $Y^*$ is trivial. Since every trivial homeomorphism has a representation, this is equivalent to conjecturing that under PFA every homeomorphism $\Phi\colon X^*\to Y^*$ between remainders of locally compact, non-compact, Polish spaces $X$ and $Y$ has a representation. The conjecture is motivated by the stated automorphism conjectures for Boolean algebras and concerns the rigidity of Stone–Čech remainders under PFA.