Hegedüs's balanced Sperner family conjecture

Let $[n]=\{1,\dots,n\}$. A family $\mathcal{F}\subseteq 2^{[n]}$ is Sperner if no two distinct members are comparable under inclusion. It is $2$-balanced if there are pairwise disjoint nonempty sets $I_1,I_2\subseteq[m]$, where $\mathcal{F}=\{F_1,\dots,F_m\}$, such that

$$\bigcup_{i\in I_1}F_i=\bigcup_{i\in I_2}F_i$$

and

$$\bigcap_{i\in I_1}F_i=\bigcap_{i\in I_2}F_i.$$

Hegedüs's conjecture. If $\mathcal{F}\subseteq 2^{[n]}$ is a Sperner family and $|\mathcal{F}|\ge n+1$, then $\mathcal{F}$ is $2$-balanced.

The conjecture was proposed after Hegedüs proved the corresponding statement for uniform families. The supplied abstract states that the result for $r=2$ has been proved with the sharp threshold $n+1$, so this conjecture is solved.