Let k be a field of characteristic zero.

Let p,q be two nonconstant polynomials in k[x,y], I the ideal generated by p and q and d an involution on k[x,y], namely, a k-algebra automorphism of order two.

Denote the set of symmetric elements with respect to d by S_d and the set of skew-symmetric elements with respect to d by K_d.

Assume that d(I) is contained in I (= I is invariant under d).

Further assume that: (i) d: (x,y) -> (x,-y). (ii) (p,q) is a Jacobian pair, namely, there exists a k-algebra endomorphism f: (x,y) -> (p,q) such that Jac(p,q):=p_xq_y-p_yq_x is in k-0.

Is the following claim true? Claim: There exist k-algebra automorphisms g,h of k[x,y], such that one of (gfh)(x),(gfh)(y) belongs to S_e or K_e, where e is an involution on k[x,y].

See https://math.stackexchange.com/questions/3350625/ideal-of-kx-y-invariant-under-an-involution

Thank you very much!