Ultraflat real Littlewood polynomials

For every $\epsilon \in (0,1)$ and every sufficiently large integer N, there is a polynomial of length N with coefficients in $\{-1,1\}$ whose modulus lies between $(1-\epsilon )\sqrt{N}$ and $(1+\epsilon )\sqrt{N}$ everywhere on the unit circle. Thus real Littlewood polynomials can be ultraflat through every sufficiently large integer length. The signs may be chosen separately at each length.