Here's a stab in the dark...

Some amps use what is called a noise gate. Basically, when the input level is very low, the output is disabled. This was done as a band aid for gain hiss, and other noises that only show at very low volumes. These were huge back in my IASCA days, since noise cost you points. From memory, Alpine was one of the first to use this technology in their amps.

So, if this circuit is present, it might not be kicking in at the normal low volume level, but it does when you turn it up. If this is the case, I don't know how to test for it, and it probably means a new amp. You could see if Alpine will admit to using a gate, for starters. Even if it has one, they might not tell you, though.

Checking the remote voltage, or better yet, jumping it from the B+ of the amp couldn't hurt, although I don't see how turning the volume up would compensate for low turn on voltage. Unless the amp also has some sort of input sensing, like for high level applications, and that is taking over.

Toby