Groebner bases for spaces of quadrics of codimension 3

Let $R=oplus_{igeq 0} R_i$ be an Artinian standard graded $K$-algebra defined by quadrics. Assume that $dim R_2leq 3$ and that $K$ is algebraically closed of characteristic $ eq 2$. We show that $R$ is defined by a Grobner basis of quadrics with, essentially, one exception. The exception is given by $K[x,y,z]/I$ where $I$ is a complete intersection of 3 quadrics not containing the square of a linear form.