Parabolic restriction dimension conjecture for large Weyl sums

For an integer $d\geqslant2$ and $\alpha\in(0,1)$, define

$$\mathcal E_{d,\alpha}=\{\mathbf x\in\mathsf{T}_d:|S_d(\mathbf x;N)|\geq N^\alpha\text{ for infinitely many }N\in\mathbb N\}.$$

Let $\mathcal X=\{(t,t^2):t\in[0,1]\}\subset\mathsf{T}_2$. Parabolic restriction dimension conjecture.

$$\dim_H(\mathcal E_{2,\alpha}\cap\mathcal X)=\max\{\dim_H\mathcal E_{2,\alpha}-1,0\}.$$

This is a conjectural dimension formula for restricting the large-values set of quadratic Weyl sums to a parabola; the source presents it as an open special case of a broader question about fractal or structured parameter sets.