On a purely inseparable analogue of the Abhyankar conjecture for affine curves

In 1957, Abhyankar proposed a conjectural description of the set of finite quotients of
Grothendieck’s  étale fundamental group for an affine curve in positive characteristic.
After a breakthrough due to Serre, in 1994, Raynaud and Harbater solved the conjecture
affirmatively. In this talk, we will discuss on a purely inseparable analogue of Abhyankar’s
conjecture, formulated in terms of Nori’s profinite fundamental group scheme. I would like
to talk about a partial answer to it.