The lower-bound conjecture for covering conical grids

Let $\Gamma$ be a conical grid of order $n$. The covering number with multiplicity $k$ is the minimum number of lines required to cover every point of $\Gamma$ at least $k$ times. Conical-grid covering conjecture. The minimum number of lines required to cover every point in $\Gamma$ at least $k$ times is at least

$$\frac{2nk}{3}-O(k).$$

The conjecture asserts that the lower bound known for structured triangular grids also holds for arbitrary conical grids, and is tight for fixed $n$ as $k\to\infty$ according to the surrounding discussion. No proof or disproof is supplied here.