Determine whether stable continuous bijections are homeomorphisms

Determine whether every stable continuous bijection from a stably compact space to a Hausdorff stable topological space is a homeomorphism, or construct a counterexample showing that this conclusion fails in the general stable setting.

Background

Corollary 5.5 establishes a partial analogue of the classical compact-to-Hausdorff theorem: if a stable continuous bijection maps a stably compact space into a Hausdorff stable topological space, then the image of every stable closed subset is stable closed.

The unresolved issue is whether the bijection must also be a homeomorphism. The authors conjecture that the answer is negative but explicitly state that they have not yet found a counterexample.

References

The prototype of Corollary \ref{corollary5.5} is that $f$ is a homeomorphism when $B={0,1}$, but we do not know whether $f$ is still a homeomorphism in the general case of Corollary \ref{corollary5.5}. We conjecture that the answer is negative, but we have not found a counterexample yet.

— Stable topology and stable compactness  (2610.22185 - Guo et al., 15 Oct 2026) in Remark 5.6, Section 5