The linear-growth conjecture for integer coverings of triangular grids

Let $f(n,d,k)$ be the minimum number of hyperplanes, counted with multiplicity, needed to cover every point of the $d$-dimensional triangular grid $T_d(n)$ at least $k$ times. Saying that the function is linear in $n$ means that its dependence on $n$ has a constant slope, up to a bounded error.

Linear-growth conjecture. For all fixed $d$ and $k$, there exists a constant $C_{d,k}$ such that

$$f(n,d,k)=C_{d,k}n+O_{d,k}(1).$$

The paper has a linear upper bound and a trivial linear lower bound, while the conjecture asserts that an asymptotic linear formula always exists. The stronger explicit formula proposed later would imply this conjecture.