  We prove the uniqueness, up to diffeomorphism, of symplectically aspherical
fillings of the unit cotangent bundle of the $3$-sphere $S^3$ under a certain
topological assumption, which Stein fillings automatically satisfy. In the
course of the proof, we show that any symplectically aspherical filling of the
unit cotangent bundle of the $n$-sphere $S^n$ ($n \geq 3$) is simply-connected.
As applications, we first show the non-existence of exact symplectic cobordisms
between some $5$-dimensional Brieskorn manifolds. We also determine the
diffeomorphism types of closed symplectic $6$-manifolds with certain
codimension $2$ symplectic submanifolds.
