Hi Terry
The first parameter that you have to fine tune is the "edge protection". This may be the most critical parameter here. A good way to find a starting point is to measure the standar deviation of the background, and use this. If you see that true edges are lost, decrease the value. Otherwise, increase it a little bit. Then, play with the strength parameter. I believe that the current default parameter is too high, so I would try it in the 1-2 range first.
If you enable the local support, the first thing to do is to decrease the strength, working in the 0.1-1 range as a starting point. By default, if you do not specify an image to be used as the local support, the luminance is used.
I'm attaching a screenshot of some parameters that worked quite well (you have to fine tune a bit further) over your jpeg.
I also would try reducing the noise in the linear stage, and see if it is easier to preserv details inside the galaxy there.