Looking for a new member of Gordon's identities

We give a commutative algebra viewpoint on Andrews recursive formula for the partitions appearing in “Gordon’s identities”, which are a generalization of Rogers-Ramanujan identities. Using this approach and differential ideals we conjecture a family of partition identities which extend Gordon’s identities. This family is indexed by r >=2. We prove the conjecture for r=2 and r=3.