Conjecture on oriented diameter of bridgeless graphs of diameter at most 3

All graphs considered are finite and simple. An orientation of a graph $G$ is obtained by replacing each edge by one of the two possible arcs with the same ends. The oriented diameter of a bridgeless graph is the minimum diameter among its orientations. A graph is bridgeless if it has no cut-edge.

Diameter-three oriented-diameter conjecture. Every bridgeless graph with diameter at most $3$ has oriented diameter at most $9$.

The paper derives the bound for several classes of spanning subgraphs and proposes the statement as the remaining general case. No resolution is supplied in the source.