It is known that for a Hausdorff locally compact etale groupoid, the reduced C*-algebra is simple iff the groupoid is minimal (meaning the orbit of each unit is dense) and topologically principal (meaning the set of units with trivial isotropy is dense).
If one considers the case where only the unit space is Hausdorff, then it is known the above two conditions are not sufficient.
Question: Are there simple to state necessary and sufficient conditions for simplicity of the reduced C*-algebra in the non-Hausdorff setting?
I suspect something can be extracted from the Khoshkam-Skandalis paper in Crelle's journal on regular representations of non-Hausdorff groupoids but I am not an operator theorist by trade.
Edit: I omitted the hypothesis that the groupoid be amenable. This should be added to both the background and the question.