[A followup on two related posts: Area of a surface confined by a sphere

Area of a elliptic surface confined by a sphere

. Thanks to all the inputs so far.]

Let $S$ be a surface enclosed inside the unit sphere in $R^3$. If

- every point of $S$ is elliptic and
- there is a point $p$ inside the unit sphere so that every half-ray emanating from $p$ intersects $S$ at most once,

then must it be the case that $\operatorname{Area}(S)\le \operatorname{Area}(S^2)$?