  A contact form on the tight $3$-sphere $(S^3,\xi_0)$ is called weakly convex
if the Conley-Zehnder index of every Reeb orbit is at least $2$. In this
article, we study Reeb flows of weakly convex contact forms on $(S^3,\xi_0)$
admitting a prescribed finite set of index-$2$ Reeb orbits, which are all
hyperbolic and mutually unlinked. We present conditions so that these index-$2$
orbits are binding orbits of a genus zero transverse foliation whose additional
binding orbits have index $3$. In addition, we show in the real-analytic case
that the topological entropy of the Reeb flow is positive if the branches of
the stable/unstable manifolds of the index-$2$ orbits are mutually
non-coincident.
