The monotonicity and high-dimensional limit conjecture for Riesz projection exponents

For $d\geq1$, let $\mathfrak{p}_d(q)$ be the critical exponent of the Riesz projection on $\mathbb{T}^d$, and let $\mathfrak{q}(d)$ be its minimal admissible exponent. Monotonicity and limit conjecture. The function $q\mapsto\mathfrak{p}_d(q)$ should be continuous and strictly increasing on $[\mathfrak{q}(d),\infty)$. Moreover,

$$\mathfrak{q}(d)\to2$$

as $d\to\infty$. This is presented as a weaker, more tangible conjecture than the full multidimensional critical exponent conjecture; the paper establishes bounds on $\mathfrak{q}(d)$ but not this assertion.