The middle even-cube covering conjecture

Let $d$ be fixed. For even $n$, call a $2d$-dimensional subcube of the hypercube a middle $2d$-cube when its layers range from $n/2-d$ to $n/2+d$. If $\mathcal A\subseteq[n]^{(n/2)}$ meets every middle $2d$-cube, the middle even-cube covering conjecture asserts

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

This generalizes the middle $4$-cube conjecture and is presented as a possible step toward showing that the limiting density for meeting $d$-cubes is $1/(d+1)$. The source identifies it with the generalized daisy conjecture for parameters $(2d,d)$.