Oriented diameter of graphs with diameter 3

In 1978, Chvátal and Thomassen proved that every 2-edge-connected graph with diameter 2 has an orientation with diameter at most 6. They also gave general bounds on the smallest value f(d) such that every 2-edge-connected graph G with diameter d has an orientation with diameter at most f(d). For d=3, their general bounds reduce to 8⩽f(3)⩽24. We improve these bounds to 9⩽f(3)⩽11.