Swinnerton-Dyer's refined Manin conjecture for cubic surfaces

Let $S'$ be a cubic surface over $\mathbb{Q}$, let $U'$ be the relevant open subset, let $H$ be its height function, and let $\rho'$ denote the corresponding Picard rank. Write $N_{U',H}(B)$ for the number of rational points of height at most $B$. Swinnerton-Dyer's refined Manin 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 $\rho'-1$, a constant $\gamma\in\mathbb{R}$, and a sequence $(\gamma_n)\subset\mathbb{C}$ such that

$$N_{U',H}(B)=Bf(\log B)+\gamma B^{\theta_3}+\mathfrak{R}e\sum_ngamma_nB^{\theta_2+it_n}+O_\epsilon(B^{\theta_1+\epsilon}).$$

Here $1/2+it_n$ runs through a set of non-trivial zeros of the Riemann zeta function, with $t_n>0$ positive and monotonically increasing, such that $\sum_n|\gamma_n|$ and $\sum_n t_n^{-2}$ converge. The conjecture refines the Manin conjecture by predicting both a more precise secondary term and an error term with power saving; the source does not report a proof or disproof, and also suggests that an analogue might hold for arbitrary surfaces.