On F-inverse covers of finite-above inverse monoids

Finite-above inverse monoids are a common generalization of finite inverse monoids and Margolis-Meakin expansions of groups. Given a finite-above E-unitary inverse monoid M and a group variety U, we find a condition for M and U, involving a construction of descending chains of graphs, which is equivalent to M having an F-inverse cover via U. In the special case where U=Ab, the variety of Abelian groups, we apply this condition to get a simple sufficient condition for M to have no F-inverse cover via Ab, formulated by means of the natural partial order and the least group congruence of M.