Quadratic Gröbner basis conjecture for binomial ideals

Let $I$ be an ideal generated by binomials, and suppose that $I$ has a quadratic Gröbner basis with respect to every monomial order. Say that the generators of $I$ are in pairwise different sets of variables when each generator involves a variable set disjoint from that of every other generator. Let $L$ be the divisor lattice of $2\cdot 3^r$, consisting of the positive divisors of $2\cdot 3^r$ ordered by divisibility. Quadratic conjecture. Either the generators of $I$ are binomials in pairwise different sets of variables, or $I=I_L$ for the divisor lattice of $2\cdot 3^r$ for some $r\geq 1$. The preceding theorem establishes the corresponding classification for finite lattices with no cut edges; this conjectural strengthening concerns arbitrary binomial ideals and is presented without a resolution.