Optimal lower bound for conical-grid multiplicity coverings

Prove that for every conical grid of order n, the minimum number of lines required to cover every point at least k times is at least 2nk − βk − O(k), where β is the constant appearing in the conjectured bound.

Background

The paper establishes a general lower bound for covering conical grids with lines and notes that the authors believe this bound is not optimal. They conjecture that a stronger lower bound, known to be tight for structured triangular grids when n and k are fixed in the relevant regime, extends to arbitrary conical grids.