Conjecture on the limiting distribution of the continuous zeta-zero orbit function

Let $S$ denote the multiset of ordinates of the nontrivial zeros of the Riemann zeta function. For $a>0$, define the continuous function $H_a:\mathbb{R}_{>0}\to\mathbb{C}$ by

$$H_a(\tau)=\frac{1}{a}\log\frac{\tau}{2\pi a}-\sum_{\substack{u\in\tau-aS\\\\ |u|\leq\pi}}(1+e^{-iu}),$$

where $\tau-aS=\\{\tau-at:t\in S\\}$ and $\lambda$ denotes Lebesgue measure. Continuous-orbit limiting-distribution conjecture. For every $a>0$ there is a Borel function $F_a:\mathbb{C}\to\mathbb{R}$ satisfying

$$\int_{\mathbb{C}}F_a=1$$

such that for every Borel set $X\subseteq\mathbb{C}$,

$$\lim_{T\to\infty}\frac{\lambda([0,T]\cap H_a^{-1}(X))}{T}=\int_XF_a.$$

The relation $\eta_{a,h}(n)=H_a((2n+1)\pi+ah)+O(n^{-1})$ connects this assertion with the discrete conjecture. Numerical evidence suggests that the statistical distribution of $H_a(\tau)$ is asymptotically independent of $\tau$, but the existence of the limiting distribution is open.