The three-dimensional covering conjecture

Let $f(n,3,k)$ be the minimum number of hyperplanes, counted with multiplicity, needed to cover every point of the three-dimensional triangular grid $T_3(n)$ at least $k$ times. The notation $O_k(1)$ denotes a quantity bounded independently of $n$ for fixed $k$.

Three-dimensional covering conjecture. For all $n,k\geq 1$,

$$f(n,3,k)= \begin{cases} \left(\frac{k+1}{2}\right)n+O_k(1)&\text{if }k\text{ is odd},\\ \left(\frac{k(k+2)}{2(k+1)}\right)n+O_k(1)&\text{if }k\text{ is even.} \end{cases}$$

The formula is supported by the proved cases $k\leq 5$, the additional asymptotic result for $k=7$, and computational evidence. It predicts linear growth in $n$ with the displayed parity-dependent slope.