Swinnerton-Dyer's refined error-term conjecture for del Pezzo surfaces

Let $S\subseteq\mathbb{P}^d$ be a possibly singular del Pezzo surface, let $U\subseteq S$ be obtained by deleting exceptional divisors, and let $\rho_S$ be the relevant Picard rank. Let $N_{U,H}(B)$ count rational points of height at most $B$. Swinnerton-Dyer's refined error-term conjecture. There exist positive constants $\theta_1,\theta_2,\theta_3<1$ with $\theta_1<\min\{\theta_2,\theta_3\}$, a polynomial $f\in\mathbb{R}[x]$ of degree $\rho_S-1$, a constant $\gamma\in\mathbb{R}$, and a sequence $(\gamma_n)\subset\mathbb{C}$ such that, for every $\varepsilon>0$,

$$N_{U,H}(B)=Bf(\log B)+\gamma B^{\theta_3}+\operatorname{Re}\sum_n\gamma_n B^{\theta_2+it_n}+O_\varepsilon(B^{\theta_1+\varepsilon}).$$

Here $\frac12+it_n$ runs through a set of non-trivial zeros of the Riemann zeta function, with positive, monotonically increasing $t_n$, and $\sum_n|\gamma_n|^2$ and $\sum_n t_n^{-2}$ convergent. The conjecture seeks a much sharper description of the error term, including oscillations arising from zeta zeros; it is presented as an open question inspired by explicit formulae in prime number theory.