The neighborhood-island conjecture for planar point sets

Let $P$ be an $n$-element set of points in the plane with no four collinear members. For $p\in P$, its neighborhood is the set of points in $P$ visible to $p$, and an island is a subset $S\subseteq P$ satisfying $\operatorname{conv}(S)\cap P=S$.

Neighborhood-island conjecture. The neighborhood of some point $p\in P$ contains an island of size $f(n)$, where $f(n)$ tends to infinity as $n$ tends to infinity.

The conjecture remains open and would imply the Big-line Big-clique Conjecture for $\ell=4$ by induction on $k$, with $n$ chosen sufficiently large.