Asymptotically minimal maxima of real Littlewood polynomials

We prove that the minimum possible maximum modulus on the unit circle of a polynomial with N consecutive real coefficients in $\{-1,1\}$ is $(1+o(1))\sqrt{N}$, as N tends to infinity through all integers. This disproves the real-sign analogue of Erdős's fixed relative-gap conjecture. As a consequence, the largest binary merit factor at length N tends to infinity through all integer lengths, disproving Turyn's conjecture.