  Let $U$ be an affine smooth curve defined over an algebraically closed field
of positive characteristic. The Abhyankar Conjecture (proved by Raynaud and
Harbater in 1994) describes the set of finite quotients of Grothendieck's
\'etale fundamental group of $U$. In this paper, we consider a purely
inseparable analogue of this problem, formulated in terms of Nori's profinite
fundamental group scheme, and give a partial answer to it.
