Sum of cosines as squashed as possible
You can get pretty good results with an approximate linear chirp spread out over the period:

$$s(x) = \sum\limits_{n=1}^{N} \cos\left(2\pi \left(x n+ \frac{n^2}{2\times N}\right)\right), \quad N=10$$

$$m \approx 4.28$$



Group delay is the derivative of the phase with respect to frequency. For a linear (frequency as function of time) chirp, the phase should be a quadratic function of frequency. With discrete frequencies we don't get exactly that, but all frequencies seem to roughly find their slot in time if we sample the phase function. I wonder if a ping pong chirp would be possible here and if it would give a flatter envelope.