Classify antimagic orientations of star forests

List all orientations of an oriented star forest S such that S admits a {0,1}-antimagic labeling, a {0,2}-antimagic labeling, and a {0,1,2}-antimagic labeling.

Background

The paper studies D-antimagic labelings of oriented stars and star forests, where the D-weight of each vertex is the sum of labels on vertices in its directed D-out-neighborhood. It establishes characterizations for individual oriented stars and for star forests made of isomorphic oriented stars, and constructs an orientation (called the Pi orientation) that works for arbitrary star forests and each of the distance sets {0,1}, {0,2}, and {0,1,2}.

The authors then exhibit an example of an antimagic oriented star forest with an orientation different from Pi. This motivates the unresolved classification problem of determining every orientation of a star forest that is simultaneously antimagic for all three specified distance sets.