Existence of a better host graph for three-in or three-out bowties

Determine whether there exists a graph G such that, for a bowtie orientation B with exactly three edges entering or exactly three edges exiting the center, D(G,B)>2^{n^2/4+1}.

Background

For the remaining bowtie orientations with exactly three incoming or three outgoing edges at the center, the authors report that the specific graph used earlier does not exceed the conjectured baseline. They also state that they have not found any graph giving a larger value, leaving open whether such a graph exists. This is a concrete unresolved existence question distinct from the subsequent conjecture asserting the exact value of D(n,B).