  Loi and Piergallini showed that a smooth compact, connected $4$-manifold $X$
with boundary admits a Stein structure if and only if $X$ is a simple branched
cover of a $4$-disk $D^4$ branched along a positive braided surface $S$ in a
bidisk $D_{1}^{2} \times D_{2}^{2} \approx D^4$. For each integer $N \geq 2$,
we construct a braided surface $S_{N}$ in $D^4$ and simple branched covers
$X_{N, 1}, X_{N, 2}, \dots , X_{N, N}$ of $D^{4}$ branched along $S_{N}$ such
that the covers have the same degrees, and they are mutually diffeomorphic, but
the Stein structures associated to the covers are mutually not homotopic.
Furthermore, by reinterpreting this result in terms of contact topology, for
each integer $N \geq 2$, we also construct a transverse link $L_{N}$ in the
standard contact $3$-sphere $(S^3, \xi_{std})$ and simple branched covers
$M_{N,1}, M_{N,2}, \ldots, M_{N, N}$ of $S^3$ branched along $L_{N}$ such that
the covers have the same degrees, and they are mutually diffeomorphic, but the
contact structures associated to the covers are mutually not isotopic.
