The explicit asymptotic formula conjecture for triangular-grid coverings

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. The parameters $q,r$ are integers subject to the displayed decompositions and ranges, and $O_{d,k}(1)$ is bounded independently of $n$ for fixed $d,k$.

Explicit asymptotic formula conjecture. If $d$ is odd and

$$k=\frac{d+1}{2}q+r,$$

with $1\leq r\leq (d+1)/2$, then

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

where

$$C_{d,k}=(q+1)\left(1+\frac{r-1}{(d+1)(q+2)/2-(r-1)}\right).$$

If $d$ is even and $k=(d+1)q+r$ with $1\leq r\leq d+1$, then

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

where

$$C_{d,k}=\begin{cases} (2q+1)\left(1+\frac{r-1}{(d+1)(q+1)-(r-1)}\right)&\text{if }1\leq r\leq d/2+1,\\ (2q+3)\left(1-\frac{d+2-r}{(d+1)(q+1)+(d+2-r)}\right)&\text{otherwise.} \end{cases}$$

This is the proposed asymptotic formula for all pairs $(d,k)$. It would in particular imply the preceding linear-growth conjecture and extend the formulas established in the paper.