Exact bowtie count for orientations with three center-directed edges

Prove that, for an orientation of the bowtie graph F_{2,3} having exactly three edges directed toward the center or exactly three edges directed away from the center, the extremal number of F_{2,3}-free orientations satisfies D(n,F_{2,3})=2^{ex(n,F_{2,3})} for sufficiently large n.

Background

The anti-directed orientations of F_{2,3} are resolved earlier in the paper, while Proposition not_semi_anti_directed_prop handles other orientations by showing that their orientation count can exceed the basic extremal-graph lower bound. The only remaining bowtie orientations are those with exactly three edges entering or exactly three edges leaving the center. The conjecture asserts that these remaining cases behave differently and attain the basic bound asymptotically.