Baernstein's norm-ratio conjecture for polynomials with unimodular roots

Let $Q(z)=1+z^n$, where $n$ is a positive integer, and let $\|\cdot\|_s$ denote the corresponding quasi-norm on the unit circle. Baernstein's conjecture. For every $0\leq s\leq t\leq\infty$ and every non-constant polynomial $p$ whose roots all lie on the unit circle,

$$\frac{\|p\|_s}{\|Q\|_s}\leq\frac{\|p\|_t}{\|Q\|_t}.$$

The paper attributes this conjecture to A. Baernstein II and explains that it would imply the entropy conjecture. Its general status is not resolved in the source.