Hardy–Littlewood discrepancy conjecture for reciprocal quadratic surds

For $n\in\mathbb{N}$ and $\theta\in\mathbb{R}\setminus\mathbb{Q}$, define

C_\theta(n):=\sum_{k=0}^n\left(\left\\{ k\theta \right\\}-\frac{1}{2}\right).

Let $a_0$ be a quadratic surd such that $a_0+1/a_0\in\mathbb{Q}$.

Hardy–Littlewood discrepancy conjecture. The function $C_{a_0}(n)+C_{1/a_0}(n)$ is bounded if and only if $a_0+1/a_0\in\mathbb{N}$. If it is bounded, it is identically zero. If it is unbounded, then it is unbounded above and below; more specifically, it is optimally $\pm\mathcal{O}(\log(n))$.

This conjecture would imply the preceding conjecture about the growth of the Ehrhart discrepancy governing infinite staircases, in the case where $a_0+1/a_0$ is not an integer. The individual Hardy–Littlewood sums are known to have logarithmic-order fluctuations, but the stated reciprocal-sum cancellation and dichotomy remain open in the paper.