Flat skew-symmetric Littlewood polynomial conjecture

Let $\mathcal L_n$ be the family of Littlewood polynomials of degree $n$. A polynomial $f(z)\in\mathcal L_n$ is skew-symmetric if its coefficients satisfy the skew-symmetry condition used in the paper. Flat skew-symmetric Littlewood polynomial conjecture. For every even $n\in\mathbb N$, there exists a skew-symmetric polynomial $f(z)\in\mathcal L_n$ and a constant $c_1>0$ independent of $n$ such that

$$|f(z)|>c_1n^{1/2}$$

on the unit circle. This is presented as a one-sided skew-symmetric version of the flat Littlewood polynomial conjecture; the general existence claim remains open.