Conjecture on the limiting distribution of the centred sums over zeta-zero cycles

Let $S$ denote the multiset of ordinates of the nontrivial zeros of the Riemann zeta function. For $a>0$, $h\in\mathbb{R}$, and $n\geq 0$, define

$$\eta_{a,h}(n)=\frac{1}{a}\log\frac{n}{a}+\sum_{\substack{t+h\in S\\\\ \left\lfloor\frac{at}{2\pi}\right\rfloor=n}}(e^{iat}-1).$$

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 $h\in\mathbb{R}$ and every Borel set $X\subseteq\mathbb{C}$,

\lim_{M\to\infty}\frac{\\#\\{0\leq n<M:\eta_{a,h}(n)+\frac{\Lambda(e^a)}{a}e^{-(\frac12+ih)a}\in X\\}}{M}=\int_Xf_a.

The preceding transformation of Landau's formula identifies the added term as the negative of the mean of $\eta_{a,h}(n)$. Numerical investigations suggest that the resulting limiting distribution is independent of $h$ and of the cycle index $n$, but the existence of the limiting Borel density remains conjectural.