Thank you for your explanation. :D
When dealing with catenative verb forms,
We are supposed to be studying.
'studying' functions as the object of 'to be'.
In terms of what syntactic role 'studying' plays, structure tells me it's a noun but semantics tells me it's a progressive verb form. I remember reading somewhere some time ago that
be + -ing functions as a pair, as a verb. Albeit, the fact that gerunds denote acts, yet function as non-acts, and, moreover, that "be" in our example sentence has a somewhat undefined ambiguity has me back at square one.
There's also the question of the subjunctive. Do such verbs take continuous verb forms as their objects? If not,
gerund seems to be the better choice.
As for nominclature, 'to be studying'
is the infinitive of "are studying", so there's no need to rename it.
I like your analysis. You've me 65% hooked. I await further evidence. :D