Butterfly supersaturation conjecture up to the packing number

Let $[n]=\{1,\ldots,n\}$, let $2^{[n]}$ denote the family of all subsets of $[n]$, and let $\Sigma(n,2)$ be the size of the two middle levels of the Boolean lattice. Write $K(n,\lceil n/2\rceil+1)$ for the largest family of $(\lceil n/2\rceil+1)$-element sets whose distinct members have intersection at most $\lceil n/2\rceil-1$, and let $f(n)$ denote the number of butterflies containing one additional set above the two middle levels.

Butterfly supersaturation conjecture. Let $E=E(n)\le K(n,\lceil n/2\rceil+1)$. If $n$ is large enough, then the minimum number of butterflies a family $\mathcal F\subset 2^{[n]}$ of size $\Sigma(n,2)+E$ must contain is $Ef(n)$.

The claim predicts the exact supersaturation threshold for families just larger than the two middle levels, within the range allowed by the packing parameter $K(n,\lceil n/2\rceil+1)$. The paper explains that its methods do not reach this range even asymptotically, so the conjecture remains open.