Conjectured edge threshold for bounded-diameter orientations of bridgeless graphs

For integers $n$ and $d$ with $2\leq d\leq n-2$, let $m(n,d)$ be the minimum number $m$ such that every bridgeless graph of order $n$ and size at least $m$ has an orientation of diameter at most $d$.

Conjecture on $m(n,d)$.

$$m(n,d)={n-d\choose 2}+n+2.$$

The formula is known for $d=2$ and $d=n-2$, while the general case remains open; the paper proves the lower bound for $5\leq d\leq n-2$.