Girth constraints for diameter-maximal regular graphs

Let $G$ be a diameter-maximal regular graph, with diameter $D$ and girth $g$.

Diameter-maximal girth conjecture. If $D$ is odd, then

$$g=D+1.$$

If $D$ is even, then

$$g\in\{D,D+1,D+2\}.$$

This conjecture is suggested by the authors' search for diameter-maximal graphs, including examples for degree $3$ and diameter at most $9$ and degree $4$ and diameter at most $6$. It is stated as an open problem.