The Tame-Wild Dichotomy for coalgebras

Let $K$ be an algebraically closed field and let $C$ be a $K$-coalgebra. The coalgebra $C$ is of tame comodule type if its finite-dimensional indecomposable comodules are parametrised by finitely many one-parameter families in each dimension, and is of wild comodule type if its comodule category contains, via an exact representation embedding, the category of finite-dimensional representations of $K\langle x,y\rangle$. Tame-Wild Dichotomy. Every $K$-coalgebra is either of tame comodule type or of wild comodule type, and these types are mutually exclusive. A weak version is known: over an algebraically closed field, tame comodule type implies not wild comodule type. The full dichotomy remains open.