Prove the optimal lower bound for conical-grid 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 − 8k − O(k).

Background

The paper proves a lower bound of approximately nk for the number of lines needed to cover every point of a conical grid at least k times. The authors state that this bound is probably not optimal and conjecture that the stronger lower bound previously obtained for structured triangular grids should hold for arbitrary conical grids. This conjectured bound is not established by the results in the paper.

References

We also believe that our lower bound on conical grids is not optimal. We claim that the lower bound in [4] for structured triangular grids, which is tight for fixed n and k > co, also holds for an arbitrary conical grid. Conjecture 2. Let IT be a conical grid of order n. The minimum number of lines required to cover every point in I at least k times is at least 2nk -Bk - O(k).

— Covering half-grids with lines and planes  (2501.11156 - Bishnoi et al., 19 Jan 2025) in Section 4, Conclusion