Kaliman–Zaidenberg rigidity conjecture for Pham–Brieskorn rings

Prove the Kaliman–Zaidenberg conjecture that, for a Pham–Brieskorn ring B_{k; a_0,\dots,a_n} over a characteristic-zero field k containing \sqrt{-1}, the ring is non-rigid if and only if some exponent a_i equals 1 or two distinct exponents a_i and a_j both equal 2.

Background

A Pham–Brieskorn ring is the quotient B_{k; a_0,\dots,a_n}=k[x_0,\dots,x_n]/(x_0^{a_0}+\cdots+x_n^{a_n}). The paper defines rigidity to mean that the only locally nilpotent derivation of the ring is the zero derivation.

The conjecture gives a complete characterization of the non-rigid cases when the base field contains \sqrt{-1}. The implication from the stated exceptional exponent patterns to non-rigidity is proved directly in the survey, and the conjecture is established for n=2 and n=3. The general higher-dimensional case remains unresolved in the paper.