On the depth of invariant rings of infinite groups

Let K be an algebraically closed field. For a finitely generated graded commutative K-algebra R, let cmdefR:=dimR−depthR denote the Cohen–Macaulay defect of R. Let G be a linear algebraic group over K that is reductive but not linearly reductive. We show that there exists a faithful rational representation V of G (which we will give explicitly) such that cmdefKG[V⊕k]⩾k−2 for all k∈N.