This is similar to what I've encountered in the past with seamingly uninspired matching of amps with speakers. Either your amp has inadequate power to drive low impedances at high volume, or your speakers are presenting difficult impedances loads (down to 1 ohm) that the amp cannot handle at even moderate volumes. Either way, I would upgrade the amp to a more powerful one that can handle difficult loads, assuming you like the sound quality of your speakers. Another possibility, however remote, is the relay protection circuitry may not be calibrated so that it shuts off at certain current levels. I've once tried bypassing a misbehaving speaker relay circuit and the amp performed well into high volumes. But I think this is rare.
I would discount the possibility of shorted speaker cables as you indicated you can listen to the amp at low volumes. A short at the speaker terminal will simply engage the protection at startup so you can't operate the amp at ANY volume.