The middle 4-cube covering conjecture

Let $n$ be even, and consider the $4$-dimensional subcubes of the hypercube whose layers range from $n/2-2$ to $n/2+2$; call these the middle 4-cubes. Let $\mathcal A\subseteq[n]^{(n/2)}$ meet every middle $4$-cube. The middle 4-cube covering conjecture.

$$|\mathcal A|\geq (1-o(1))\binom{n}{n/2}.$$

The conjecture asserts that asymptotically almost every point of the middle layer is necessary. The source describes it as a first step toward proving that the limiting density for meeting $4$-cubes is $1/5$, and relates it equivalently to the daisy problem at uniformity $n/2$.