Groebner bases for spaces of quadrics of codimension 3

Let $R={\oplus }_{i\ge 0}{R}_i$ be an Artinian standard graded $K$-algebra defined by quadrics. Assume that $\dim R_2\le 3$ and that $K$ is algebraically closed of characteristic $\mathrm{\ne }2$. We show that $R$ is defined by a Gr"obner basis of quadrics with, essentially, one exception. The exception is given by $K[x,y,z]\mathrm{/}I$ where $I$ is a complete intersection of 3 quadrics not containing the square of a linear form.