Growth conditions for a family of ideals containing regular sequences

It has been conjectured by Eisenbud–Green–Harris that lex-plus-powers ideals exhibit extremal conditions among all homogeneous ideals containing a regular sequence of forms in fixed degrees. In the same spirit, we consider a family of homogeneous ideals in k[x,y,z] which contain a regular sequence of forms F,Gset membership, variantk[x,y] and compare the growth of these ideals with special monomial ideals sharing similar properties.