The planar strip occupancy conjecture

Let $N$ be a positive integer and let $x_1,\ldots,x_N$ be points in the unit disk in the plane. A strip of width $1/N$ is a planar strip with that width. Planar strip occupancy conjecture. There is a function $g$ on the positive integers tending to infinity such that, for any $N$ points in the unit disk on the plane, there is a strip of width $1/N$ containing at least $g(N)$ of the given points. This is the planar analogue of the spherical strip question, asking for an unbounded lower bound on the number of points captured by a strip of reciprocal width; the source poses it as an open question.