The identity conjecture for compactly supported symplectomorphisms of flexible Weinstein structures

A flexible Weinstein structure is a Weinstein structure whose underlying Weinstein domain is flexible, and a compactly supported symplectomorphism is a symplectomorphism equal to the identity outside a compact subset. Identity conjecture. Any compactly supported symplectomorphism of a flexible Weinstein structure is isotopic/homotopic to the identity.

This conjecture would strengthen the preceding cohomological restriction, which shows only that an exact compactly supported symplectomorphism acts trivially on the cohomology of the flexible Weinstein domain. Whether all such symplectomorphisms are isotopic or homotopic to the identity remains open.