I've had 2 sets of HRE's. I believe that HRE says to only use soap and warm water with 100% cotton towel. They also say that you can use a paste wax with no abrasives to help in keeping them clean.
The high polish is a ***** to keep the water spots and grime off of. Mine developed the same thing as yours.
The set I have on order now will be in full chrome.IMO the only way to go from now on.