Liouville homotopy to Weinstein domains

Determine whether every Liouville domain constructed from a compact cooriented partially hyperbolic bi-contact plug as in Theorem 1.7 is homotopic, through Liouville domains, to a Weinstein Liouville domain.

Background

The paper constructs Liouville structures on thickenings of cooriented partially hyperbolic bi-contact plugs. For repelling plugs, rounding the corners produces Liouville domains whose skeletons project homeomorphically onto the plug’s maximal invariant set; in many examples the resulting Liouville dynamics are chaotic.

The authors note that the existence of Liouville domains with the indicated topological and dynamical features that are not Liouville homotopic to Weinstein domains is a well-known open problem in symplectic topology. They then pose the specific unresolved question of whether all Liouville domains arising from their construction are nevertheless homotopic through Liouville domains to some Weinstein domain.