Koszul filtration conjecture for quadratic binomial Gröbner bases

Let $S$ be a polynomial ring and let $G$ be a reduced Gröbner basis of quadratic binomials, each of whose two terms has disjoint support. A Koszul filtration is a family of ideals generated by linear forms satisfying the usual colon-ideal conditions, and a monomial Koszul filtration is one generated by subsets of the variables. If $(G)$ is toric, meaning that it is a toric ideal, then the quotient $S/(G)$ is considered with its induced grading.

Koszul filtration conjecture. The quotient $S/(G)$ has a Koszul filtration. Moreover, if $(G)$ is toric, then $S/(G)$ has a monomial Koszul filtration.

This conjecture extends the filtration results established in the paper for specific classes of quadratic binomial Gröbner bases. The supplied text does not give a resolution, so the general assertion remains open.