The multidimensional critical exponent conjecture for the Riesz projection

For $d\geq1$ and $1\leq q\leq\infty$, let $\mathfrak{p}_d(q)$ denote the critical exponent of the Riesz projection from $L^q(\mathbb{T}^d)$ to $H^p(\mathbb{T}^d)$, and define

$$\mathfrak{a}_d(q)=2+\cfrac{2}{d+\cfrac{2}{q-2}}.$$

Critical exponent conjecture. One should have

$$\mathfrak{p}_d(q)=\begin{cases}\mathfrak{a}_d(q), & \frac{2d}{d+1}\leq q\leq\infty;\\ -1, & 1\leq q<\frac{2d}{d+1}. \end{cases}$$

This extends the one-dimensional conjecture and is motivated by the composition relation for the proposed exponents. The paper provides upper and lower bounds, but the exact formula remains open.