The bowtie extremal-orientation conjecture

Let $B=F_{2,3}$ be the bowtie graph, with a distinguished center vertex. Let $G$ be a complete tripartite graph with part sizes $1$, $\left\lfloor\frac{n-1}{2}\right\rfloor$, and $\left\lceil\frac{n-1}{2}\right\rceil$. Let $\vec{B}$ be an orientation of $B$ as in Case (a) or (b) of the proposition preceding this conjecture. The bowtie extremal-orientation conjecture. For every such orientation,

$$D(G,\vec{B})=D(n,\vec{B}).$$

This conjecture identifies a complete tripartite graph as an extremal host for the specified unresolved orientations of the bowtie; the preceding proposition shows that these orientations have more than $2^{\mathrm{ex}(n,B)}$ orientations in general, while the exact value remains open.