Quadratic Gorenstein rings and the Koszul property I

Let $R$ be a standard graded Gorenstein algebra over a field presented by quadrics. Conca, Rossi, and Valla have shown that such a ring is Koszul if $\mathrm{r}\mathrm{e}\mathrm{g}R\le 2$ or if $\mathrm{r}\mathrm{e}\mathrm{g}=3$ and $c= $\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{i}\mathrm{m}$, R \leq 4$,andtheyaskwhetherthisistruefor$$\mathrm{r}\mathrm{e}\mathrm{g}$, R = 3$ in general. We determine sufficient conditions on a non-Koszul quadratic Cohen-Macaulay ring $R$ that guarantee the Nagata idealization $\tilde{R} = R \ltimes \omega_R(-a-1)$ is a non-Koszul quadratic Gorenstein ring. We use this to negatively answer the question of Conca-Rossi-Valla, constructing non-Koszul quadratic Gorenstein rings of regularity 3 for all $c\ge 9$.