Maximum-size biconnected digraphs of given radius

Let $r\geq \frac52$ and $n\geq 2r+1$. Let $\overline{\Gamma}_{2r+1}$ be the digraph specified in the paper, and let $\overline{\Gamma}_{n,2r,i,s}$ denote the family obtained by the indicated blow-ups, where $1\leq i\leq 2r-2$ and $1\leq s\leq n-2r$. Radius extremal-structure conjecture. The biconnected digraphs of order $n$ and radius $r$ maximizing the size are formed by taking two blow-ups at consecutive, non-end vertices of $\overline{\Gamma}_{2r+1}$. Furthermore, the extremal digraphs are exactly those of the form $\overline{\Gamma}_{n,2r,i,s}$ where $1\leq i\leq 2r-2$ and $1\leq s\leq n-2r$. This is presented as the radius analogue of the outradius conjecture; the small-order cases can have no simple uniform structure, whereas the stated range is conjectural.