Bau–Dankelmann oriented diameter conjecture for bridgeless graphs

Let $G$ be a bridgeless graph of order $n$ and minimum degree $\delta\geq 2$, and let $\overrightarrow{\operatorname{diam}}(G)$ denote the minimum diameter over all strongly connected orientations of $G$. Bau–Dankelmann's oriented diameter conjecture. There exists a constant $C$ such that

$$\overrightarrow{\operatorname{diam}}(G) \leq \frac{3n}{\delta+1}+C.$$

This conjecture asks for the asymptotically sharp upper bound suggested by the known lower bounds for bridgeless graphs. The paper proves an asymptotic result with coefficient $3+\epsilon$ and denominator $\delta-2$, while the conjectured bound with denominator $\delta+1$ and an additive constant remains open.