Converse separation: generically c.e. but not coarsely c.e. at level Σα+2

Determine whether there exists a countable linear ordering L and a computable ordinal α such that L is Σα+2-generically computably enumerable (c.e.) but not Σα+2-coarsely c.e.

Background

The paper proves that for every computable ordinal α, there are linear orderings that are Σα+2-coarsely c.e. but not Σα+2-generically c.e. (Proposition 4.2), establishing a one-directional separation between generic and coarse computability at arbitrarily high levels.

The authors explicitly leave open the converse direction: whether there exist linear orderings that are Σα+2-generically c.e. yet fail to be Σα+2-coarsely c.e., which would demonstrate a strict non-coincidence of these notions at the same level from the opposite direction.