Working with screen's touch limit

I have a simple game that contains four characters the player can touch, on a large touch screen running Windows that supports up to 12 simultaneous touches. Currently I loop through Input.touches and raycast each frame, but unfortunately it's possible that all 12 touch points have been 'spent' somewhere else on the screen that frame.

Eg. one person puts all 10 of their fingers (and perhaps their palms) on the screen to 'use up' all 12 touch points, and then a genuine player tries to touch the four characters and these touches are simply ignored.

I'm aware this is likely a hardware limitation I can't do anything about, but does anybody know if there's anything we can do to ensure the four touchable items still respond? I'd initially hoped it were possible to ignore touches that weren't in the 'start' phase so that it was much harder to use up all 12 touches but this doesn't seem possible either.