Complexity for large discrete witness sets

Determine the exact computational complexity of the Discrete Witness Set Problem in weak visibility polygons when the number of input points m is larger than the number of polygon vertices n.

Background

The paper gives an optimal $\Theta(n\log n)$ bound when the number of input points satisfies $m=\Theta(n)$. However, every witness set in a weak visibility polygon has at most $\rho+1$ members, where $\rho$ is the number of reflex vertices, so when $m$ is much larger than $n$, most input points cannot belong to an optimum solution.

The authors explicitly leave unresolved the resulting complexity of the discrete problem in this larger-input regime.