  We prove that all flexible Weinstein fillings of a given contact manifold
with vanishing first Chern class have isomorphic integral cohomology; in
certain cases, we prove that all flexible fillings are symplectomorphic. As an
application, we show that in dimension at least 5 any almost contact class that
has an almost Weinstein filling has infinitely many different contact
structures. Similar methods are used to construct the first known infinite
family of almost symplectomorphic Weinstein domains whose contact boundaries
are not contactomorphic. We also prove relative analogs of our results, which
we apply to Lagrangians in cotangent bundles.
