The restricted almost-cover conjecture for hypercubes

Let $Q^n=[0,1]^n$ be the $n$-dimensional hypercube. Fix $k\geq 1$ and sufficiently large $n$, and let $H_1,\ldots,H_m$ be affine hyperplanes in $\mathbb{R}^n$ not containing $\vec{0}$. Suppose that every vector in $Q^n$ with exactly $t$ coordinates equal to $1$ is covered by these hyperplanes at least $k-t$ times, for each $t=1,\ldots,k-1$.

Restricted almost-cover conjecture. Under these assumptions,

$$m\geq\binom{k}{2}.$$

This is presented as a weaker consequence of the main almost-cover conjecture when the cover is restricted to use the coordinate hyperplanes $x_i=1$. It remains open in the stated generality.