The two-dimensional integer covering asymptotic conjecture

Let $f(n,2,k)$ be the minimum number of lines, counted with multiplicity, that cover every point of the triangular grid $T_2(n)$ at least $k$ times, and let $f^*(k,2)$ be the fractional covering density for multiplicity $k$ on $T_2(k)$. The notation $O_k(1)$ denotes a quantity bounded independently of $n$ for fixed $k$.

Two-dimensional asymptotic conjecture. For all positive integers $n$ and $k$,

$$f(n,2,k)=f^*(k,2)n+O_k(1).$$

Computational evidence for $k\leq 7$ motivates the conjecture, which would identify the fractional covering density as the asymptotic slope of the integer problem in dimension two.