Weyl-sum orbit density conjecture

For an integer $d\geqslant2$ and $\mathbf x\in\mathsf{T}_d$, let

$$O_d(\mathbf x)=\{S_d(\mathbf x;N):N\in\mathbb N\}$$

be the orbit of the Weyl sums. Weyl-sum orbit density conjecture. For almost all $\mathbf x\in\mathsf{T}_d$ in the sense of Lebesgue measure, the orbit $O_d(\mathbf x)$ is everywhere dense in $\mathbb C$. The conjecture is motivated by a random-walk model for Weyl sums, while the source also notes that particular rational or badly approximable parameters can have nondense or bounded orbits.