The aperiodic monotile problem asks whether there exists a single tile that every tiling of the plane made with it results non-periodic. What is known about this problem? If this tile exists, how can it be/not be? how could it (not) be constructed?

EDIT: I'm thinking about the most restrictive case where the single tile should be homeomorphic to an euclidean closed disk.

EDIT: The kind of answer I'm looking for is, for example, this result implies that if the monotile is to exist, it cannot be convex.