Contractive Riesz projection conjecture and its endpoint logarithmic inequality

Let $P$ denote the Riesz projection from $L^2(\mathbb{T})$ onto $H^2$, and let $r$ and $q$ satisfy

$$1<r\leq\infty,\qquad q=4\left(1-\frac{1}{r}\right).$$

Riesz projection contractivity conjecture. The operator $P$ is a contraction from $L^r(\mathbb{T})$ to $H^q$. In addition, for every $f\in L^1(\mathbb{T})$,

$$\exp\left(\int_0^{2\pi}\log|Pf(e^{i\theta})|\,\frac{d\theta}{2\pi}\right)\leq\|f\|_{L^1}.$$

This conjecture connects the coefficient inequalities with contractive bounds for the Riesz projection. The source notes the endpoint $L^\infty(\mathbb{T})\to H^4$ result and its sharpness, but does not resolve the full asserted family or the accompanying $L^1$ inequality.