is whether or not the attack level will ALWAYS eventually go up to the maximum, or does it's max level depend on the peak voltage of the trigger pulse?
No there is nothing that suggests it will always go up to a maximum, it is DC coupled throughout so the output is eventually going to be function of the input.
To have it always increase I would expect to see a capacitor in the feed back loop of one or more amplifier, in other words an integrator.