On partial clones containing all idempotent partial operations

Let k>2 and k be a k-element set. We show that though the partial clone I:=∩_a∈kpPol {a} of all idempotent partial operations on k is contained in finitely many partial clones on k, it is quite hard to determine the number of all partial clones on k containing the partial clone I on k. Indeed this question is related to a hard problem in the area of extremal set theory