Generalized Abhyankar conjecture for the affine line

Let $k$ be a field of characteristic $p>5$, and let $\mathbb{A}^1_k$ be the affine line over $k$. Let $\pi^{\rm loc}_A(\mathbb{A}^1_k)$ denote the set of finite local $k$-group schemes appearing as quotients of the local fundamental group scheme of $\mathbb{A}^1_k$. A finite local $k$-group scheme is said to have no nontrivial characters when it admits no nontrivial homomorphism to the multiplicative group scheme. Generalized Abhyankar conjecture. The set $\pi^{\rm loc}_A(\mathbb{A}^1_k)$ is the set of finite local $k$-group schemes with no nontrivial characters. This is the affine-line formulation of the purely inseparable analogue of Abhyankar's conjecture; the source presents it as a question, and the general assertion remains open, although evidence and the solvable case are known.