Let $k$ be a field and $A$ be a noncommutative $k$-algebra with Jacobson radical $J$. If $A$ is finite-dimensional, the Wedderburn-Malcev says that $A$ has a subalgebra $S$ such that $$A = S \oplus J$$ as vector spaces. Algebras with such a subalgebra $S$ are sometimes called cleft, and $S$ is said to be a cleaving of $A$.
Curtis generalized this to some infinite-dimensional algebras: if $\cap_n J^n = 0$ and $A$ is complete with respect to the $J$-adic topology, then $A$ is cleft.
I am looking for a generalization of the Wedderburn-Malcev theorem to infinite-dimensional, graded-finite algebras.
Consider the following example not covered by Curtis' result, but for which the conclusion still holds: if $A= k[x]$ has its usual grading, then $A$ has a cleaving, namely $k[x]= k \oplus (x)$.
Let $A$ be a $\mathbb{Z}$-graded $k$-algebra which is graded-finite. Let $J$ be its (graded) Jacobson radical. Does $A$ have a subalgebra $S$ such that $A = S\oplus J$?
Edit: if it helps, we can also assume that $A$ lives in degrees above $l$ for some $l\in \mathbb{Z}$.
