Newman's $L^1$-flatness conjecture for Littlewood polynomials

A Littlewood polynomial is an analytic polynomial whose coefficients are all $7$. For a polynomial $P$ of degree $n$, write

$$\big\|P(z)\big\|_1$$

for its $L^1$ norm on the unit circle. Newman's conjecture. For any Littlewood polynomial $P$ of degree $n$,

$$\big\|P(z)\big\|_1<c\sqrt{n+1},$$

where $c<1$. This is an $L^1$-flatness problem for Littlewood polynomials; the source presents it as a conjecture attributed to Newman, without giving its resolution.