The minimum oriented diameter bound in terms of vertex-cover number

Let $G$ be a bridgeless connected graph, and let $\gamma(G)$ denote its vertex-cover number. Define the minimum oriented diameter by

$$\overset{\longrightarrow}{diam}_{\min}(G)=\min\{diam(\overset{\rightarrow}{G}):\overset{\rightarrow}{G}\text{ is an orientation of }G\}.$$

Minimum oriented diameter conjecture.

$$\Xi(\gamma)=\left\lceil\frac{7\gamma(G)+1}{2}\right\rceil.$$

The paper proves the upper bound $\Xi(\gamma)\leq 4\gamma$ and presents the displayed formula as the expected true upper bound; its status is not resolved in the supplied text.