The Nowicki conjecture for relatively free algebras

A linear locally nilpotent derivation of the polynomial algebra K[Xm] in m variables over a field K of characteristic 0 is called a Weitzenböck derivation. It is well known from the classical theorem of Weitzenböck that the algebra of constants K[Xm]δ of a Weitzenböck derivation δ is finitely generated. Assume that δ acts on the polynomial algebra K[X2d] in 2d variables as follows: δ(x2i)=x2i−1, δ(x2i−1)=0, i=1,…,d. The Nowicki conjecture states that the algebra K[X2d]δ is generated by x1,x3.…,x2d−1, and x2i−1x2j−x2ix2j−1, 1≤