The asymptotic almost-cover conjecture for hypercubes

Let $Q^n=[0,1]^n$ be the $n$-dimensional hypercube, and let $f(n,k)$ denote the minimum number of affine hyperplanes whose collection is an almost $k$-cover of $Q^n$, meaning that every vertex of $Q^n$ is covered at least $k$ times except possibly the origin.

Almost-cover conjecture. For an arbitrary fixed integer $k 1$ and sufficiently large $n$,

$$f(n,k)=n+\binom{k}{2}.$$

Equivalently, for large $n$, an almost $k$-cover of $Q^n$ contains at least $n+\binom{k}{2}$ affine hyperplanes. This is proposed for larger $k$ after the exact values are established for $k\leq 3$; the conjecture remains open in general, with the paper noting the case $k=4$ as a particular example.