Large balanced-subset-free families of middle subsets

For positive integers $k$ and sufficiently large integers $n$, let $\mathcal{F}$ be a family of $n$-subsets of $[2n]$. A subset of members of $\mathcal{F}$ is balanced when it has the balancing property used in the definition of the frame index.

Balanced-subset family conjecture. For each $k$ and all sufficiently large $n$, there is a family $\mathcal{F}$ of $n$-subsets of $[2n]$ such that:

1. $\mathcal{F}$ has no balanced subsets of sizes $2,4,\ldots,2k$.
2. $\mathcal{F}$ has a balanced subset of size $2k+2$.
3. $\mathcal{F}$ is as large as possible subject to these properties, and therefore

$$\lvert\mathcal{F}\rvert=\frac{1}{2}\binom{2n}{n}.$$

This conjecture generalizes the stated construction for index $6$ to larger even indices by allowing the ground-set parameter $n$ to grow. The existence of such families for every $k$ and all sufficiently large $n$ remains open.