The problem with a water softener is that it only temporarily lowers the ph. When I lived in my parents' house for a short time before my wife and I got our own house, I tested the ph from their tap and it came out at about 7.0. However, after it sat for about 24 hours, it was up at about 8.2-8.4. This creates a problem for fish because the ph in the tank is that of aged water, which is about 8.4. However, when a water change is done, say 50%, it will drop the ph from 8.4 to 7.7, immediately, and shock fish that are sensitive to it. Then, 24 hours later the ph will be back up at 8.4.
If you want to run a water softener then I recommend setting up some sort of water aging container. That way you can give the water a change to level out before adding it to the tank.