Hamaker–Pechenik–Weigandt's multiplicity-free Schubert polynomial conjecture

Let $w\in S_n$ be a permutation, and let its single Schubert polynomial be a sum of monomials. A sum of monomials is multiplicity-free when every monomial occurs with coefficient one.

Hamaker–Pechenik–Weigandt's conjecture. If the Schubert polynomial of $w$ is a multiplicity-free sum of monomials, then the CDG generators of $I_w$ are a diagonal Gröbner basis.

This conjecture connects multiplicity-freeness of Schubert polynomials with the Gröbner-basis property of CDG generators. The source says that the first conjecture implies this one, but does not state here that this consequence has itself been proved; its status is therefore left open.