The diameter conjecture on flat tori in $S^3$ states that the extrinsic diameter of every flat torus isometrically immersed in the unit 3-sphere $S^3$ is equal to $\pi$. This conjecture is closely related to the rigidity problem on the Clifford tori in $S^3$. In this talk, introducing a representation theorem for flat tori in $S^3$, I explain a recent result which shows that the conjecture is true for every isometrically immersed flat torus in $S^3$ whose mean curvature function does not change sign. This is a joint work with Masaaki Umehara (Osaka).