Brevig–Ortega-Cerdà–Seip–Zhao conjecture on the Riesz projection

For $1\leq p<\infty$, let $p'=\frac{p}{p-1}$, let $P_+$ denote the Riesz projection on $L^{p'}(\mathbb T)$, and let $H^{4/p}$ be the Hardy space with exponent $4/p$. Brevig–Ortega-Cerdà–Seip–Zhao conjecture. The operator $P_+$ is a contraction from $L^{p'}(\mathbb T)$ to $H^{4/p}$. In the endpoint case $p=\infty$, the corresponding assertion is

$$\exp\left(\frac{1}{2\pi}\int_0^{2\pi}\log\left|P_+F(e^{it})\right|\,dt\right)\leq\|F\|_{L^1}$$

for every $F\in L^1(\mathbb T)$. The source presents this as an unresolved conjecture concerning the norm of the Riesz projection into Hardy spaces, following known contractive-range results and an endpoint inequality.