Parabolic natural-measure conjecture for large quadratic Weyl sums

Define

$$\mathcal E_{2,1/2}=\{\mathbf x\in\mathsf{T}_2:|S_2(\mathbf x;N)|\geq N^{1/2}\text{ for infinitely many }N\in\mathbb N\}.$$

Let $\mathcal X=\{(t,t^2):t\in[0,1]\}$ and let $\mu$ be the natural probability measure on $\mathcal X$. Parabolic natural-measure conjecture.

$$\mu(\mathcal E_{2,1/2}\cap\mathcal X)=1.$$

This asserts that the large-values property at the critical exponent holds for almost every point of the parabola with respect to its natural measure; the source supplies no resolution.