Let's consider a somewhat similar case.

You are an inventor. An evil dictator captures you, and takes you off to a faraway dungeon, where he tells you that he wants you to build him a superweapon. If you refuse to build the weapon, well, he has means of persuading you. If you still refuse, he will kill you.

Of course, when you finish building the doomsday device, your usefulness will be over, and he will probably kill you anyway.

Being a genius, you soon realise that you could build the dictator a doomsday device in about a week, with the materials lying around in your well-equipped dungeon workshop. However, this doesn't seem terribly optimal.

What do you do? You agree to build the doomsday device. But you claim it is very difficult. You spend a long time producing impressive prototypes that seem to suggest you're making progress. You claim that you need more resources. You carry on, making just enough apparent progress to stop him from killing you on the spot, but not enough to actually succeed. You build ever more expensive and complex non-working prototypes, hoping that in the course of this, you can either build yourself something suitable for breaking out of the dungeon or killing the evil dictator. Or hoping that perhaps someone will rescue you. At the very least you will have wasted the evil dictator's resources on doing something pointless - you will at least not die for nothing.

I suspect your imprisoned AI may choose to follow a similar course.