The projective-dimension-two conjecture for monomial ideals

Let $M$ be a monomial ideal minimally generated by more than one monomial. Suppose that some minimal generator divides the least common multiple of every pair of generators. Projective-dimension-two conjecture. Then

$$\operatorname{pd}(S/M)=2.$$

This is proposed as a simple generalization of an earlier theorem on the projective dimension of monomial ideals. The source does not report a proof or disproof.