On the geometry of zero sets of central quaternionic polynomials

Let R be the ring of polynomials in n central variables over the real quaternion algebra H, and let I be a left ideal in R. We prove that if a polynomial p in R vanishes at all the common zeros of I in H^n with commuting coordinates, then as a slice regular quaternionic function, p vanishes at all common zeros of I in H^n. This confirms a conjecture of Gori, Sarfatti and Vlacci, who settled the two dimensional case.