  A question of Conca, Rossi, and Valla asks whether every quadratic Gorenstein
ring $R$ of regularity three is Koszul. In a previous paper, we use
idealization to answer their question, proving that in nine or more variables
there exist quadratic Gorenstein rings of regularity three which are not
Koszul. In this paper, we study the analog of the Conca-Rossi-Valla question
when the regularity of $R$ is four or more. Let $R$ be a quadratic Gorenstein
ring having $\mathrm{codim}\, R = c$ and $\mathrm{reg}\, R = r \ge 4$. We prove
that if $c = r+1$ then $R$ is always Koszul, and for every $c \geq r+2$, we
construct quadratic Gorenstein rings that are not Koszul, answering questions
of Matsuda and Migliore-Nagel concerning the $h$-vectors of quadratic
Gorenstein rings.
