The extremizer conjecture for the critical Riesz projection inequality

Let $1<q<2$ or $2<q\leq\infty$, let $\mathfrak{p}(q)$ be the one-dimensional critical exponent, and let $P_+$ denote the Riesz projection on the unit circle. Extremizer conjecture. If a nontrivial function $\psi\in L^q$ satisfies

$$\|P_+\psi\|_{\mathfrak{p}(q)}=\|\psi\|_q,$$

then there are a constant $C\neq0$ and an inner function $I$ such that $\psi=CI$. The endpoint $q=1$ is described as exceptional, and the asserted characterization is conjectured for the stated non-endpoint ranges.