  Let $\mathfrak{p}_d(q)$ denote the critical exponent of the Riesz projection
from $L^q(\mathbb{T}^d)$ to the Hardy space $H^p(\mathbb{T}^d)$, where
$\mathbb{T}$ is the unit circle. We present the state-of-the-art on the
conjecture that $\mathfrak{p}_1(q) = 4(1-1/q)$ for $1 \leq q \leq \infty$ and
prove that it holds in the endpoint case $q = 1$. We then extend the conjecture
to \[\mathfrak{p}_d(q) = 2+\cfrac{2}{d+\cfrac{2}{q-2}}\] for $d\geq1$ and
$\frac{2d}{d+1} \leq q \leq \infty$ and establish that if the conjecture holds
for $d=1$, then it also holds for $d=2$. When $d=2$, we verify that the
conjecture holds in the endpoint case $q = 4/3$.
