On the maxima of Littlewood polynomials on $[-1,1]$

A Littlewood polynomial is a polynomial of the form $$f_n(x)=\sum_{k=0}^n \varepsilon_k x^k$$ with $\varepsilon_k\in\{-1, 1\}$. Let $(\varepsilon_k)_{k \ge 0}$ be i.i.d. Rademacher coefficients. We show that the lower envelope of $\max_{x\in[-1,1]}|f_n(x)|$ is determined by the small-ball probability of a certain Gaussian process. In particular, almost surely, $$\liminf_{n\to\infty} \frac{\log(\max_{x\in[-1,1]}|f_n(x)|/\sqrt n)}{(\log\log n)^{1/3}} = -\Big(\frac{3\pi^2}{4}\Big)^{1/3}.$$