Exact value for the remaining three-in or three-out bowtie orientations

Establish that, for every orientation B of the bowtie graph F_{2,3} with exactly three edges directed toward the center or exactly three edges directed away from the center, D(n,B)=2^{ex(n,B)} for sufficiently large n.

Background

The anti-directed bowtie orientations are handled by Theorem \ref{k-fans_thm_1}, while Proposition \ref{not_semi_anti_directed_prop} handles other orientations and shows that some have more than 2^{ex(n,B)} free orientations. The only remaining bowtie orientations are those with exactly three edges entering or leaving the center. The conjecture asserts that these exceptional orientations still satisfy the basic extremal equality.