Batyrev and Popov's quadratic generation conjecture for Cox rings of del Pezzo surfaces

Let $X$ be a del Pezzo surface. Let $I_X$ be the defining ideal in a presentation ${\rm Cox}(X)\cong k[\mathcal G]/I_X$, where $\mathcal G$ is a minimal homogeneous generating set of the Cox ring. Batyrev and Popov's conjecture. The ideal $I_X$ is generated by quadrics. This conjecture predicts a uniform simple presentation for the Cox rings of del Pezzo surfaces; the supplied source does not indicate whether it has been resolved in this generality.