The three-cover multiplicity conjecture for hypercubes

Let $Q^n=\{0,1\}^n$ be the vertex set of the $n$-dimensional hypercube, and let $g(n,m,k)$ be the minimum number of vertices covered fewer than $k$ times by $m$ affine hyperplanes not passing through $\vec{0}$. For $n\geq m\geq 1$, the conjecture concerns the range $1\leq m\leq n+3$.

Three-cover multiplicity conjecture.

$$g(n,m,3)= \begin{cases} 2^n, & m=1,2;\\ 2^{n-1}, & m=3;\\ 2^{n-m+3}, & m=4,\ldots,n+3. \end{cases}$$

The first two cases and the stated lower and upper bounds motivate the conjecture; the paper does not report a proof of the full piecewise formula, so it remains open.