  For any asymptotically dynamically convex contact manifold $Y$, we show that
$SH_*(W)=0$ is a property independent of the choice of topologically simple
(i.e.\ $c_1(W)=0$ and $\pi_{1}(Y)\rightarrow \pi_1(W)$ is injective) Liouville
filling $W$. In particular, if $Y$ is the boundary of a flexible Weinstein
domain, then any topologically simple Liouville filling $W$ has vanishing
symplectic homology. As a consequence, we answer a question of Lazarev
partially: a contact manifold $Y$ admitting flexible fillings determines the
integral cohomology of all the topologically simple Liouville fillings of $Y$.
The vanishing result provides an obstruction to flexible fillability. As an
application, we show that all Brieskorn manifolds of dimension $\ge 5$ cannot
be filled by flexible Weinstein manifolds.
