Butterfly trace conjecture

Let $B$ be the butterfly poset on four elements $a,b,c,d$, with $a,b<_B c,d$. For a family $\mathcal F\subseteq 2^{[n]}$, let $Tr_{n-1}(n,B)$ denote the maximum size of a family whose traces on all $(n-1)$-subsets contain no copy of $B$. Butterfly trace conjecture. If $n\geq 5$, then

$$Tr_{n-1}(n,B)=\binom{n}{\lfloor n/2\rfloor}.$$

The paper proves the related exact formula $Tr(n,B)=\lfloor 3n/2\rfloor+1$, but the asserted value for traces on $(n-1)$-subsets is posed as an open problem.