Existence of extremal graphs exceeding the standard bound for three-in or three-out bowties

Determine whether there exists an n-vertex graph G with D(G,B)>2^{n^2/4+1} for an orientation B of the bowtie graph F_{2,3} having exactly three edges entering the center vertex or exactly three edges leaving the center vertex.

Background

After proposing the conjecture that the three-in and three-out bowtie orientations satisfy the standard extremal equality, the authors report that the natural candidate graph from the preceding proposition does not produce enough B-free orientations.

They explicitly state that they have not found any graph exceeding the bound, leaving the existence of such a graph unresolved. This is a concrete unresolved existence question distinct from the conjectured exact value.

References

When this is the case, one can easily check that the graph $G$ from the proof of Proposition \ref{not_semi_anti_directed_prop} has fewer than $2{n2/4}+1}$ many $B$-free orientations. Indeed, we have been unable to find any graph $G$ such that $D(G, B)>2{n2/4}+1}$.

— Orientations of graphs omitting non-edge-critical directed graphs  (2502.21287 - Sheats, 28 Feb 2025) in Section 5, subsection “Other orientations of F_{2,3} and examples,” paragraph immediately preceding the discussion of wheel graphs