  We consider symplectic cohomology twisted by sphere bundles, which can be
viewed as an analogue of local systems. Using the associated Gysin exact
sequence, we prove the uniqueness of part of the ring structure on cohomology
of fillings for those asymptotically dynamically convex manifolds with
vanishing property considered in [30,31]. In particular, for simply connected
$4n+1$ dimensional flexible fillable contact $Y$, we show that real cohomology
$H^*(W)$ is unique as a ring for any Liouville filling $W$ of $Y$ as long as
$c_1(W)=0$. Uniqueness of real homotopy type of Liouville fillings is also
obtained for a class of flexibly fillable contact manifolds.
