The variable-width spherical strip occupancy conjecture

Let $N$ be a positive integer, let $x_1,\ldots,x_N$ be points on the unit sphere $\mathbb S^{2}$, and let $w>0$. For a unit vector $v$, consider the strip $|\langle v,x\rangle|\leq w$. Variable-width spherical strip occupancy conjecture. There is a function $h$ on the positive integers tending to infinity such that, for any $N$ points on $\mathbb S^{2}$ and any width $w>0$, there is a unit vector $v$ for which either at least $h(N)$ given points lie in the strip $|\langle v,x\rangle|\leq w$, or no chosen point lies in the strip $|\langle v,x\rangle|\leq w$. This is a stronger-looking alternative formulation allowing arbitrary positive strip width, with the empty-strip alternative accounting for configurations and widths where a populated strip need not be available; the source leaves the question open.