Can a Boggart "see" through the Hallows (Harry's) cloak?

Question

In Harry Potter and the Prisoner of Azkaban, Professor Lupin states that nobody can see the true form of a Boggart and that it is best to take two or more people with you to deal with one.

What would happen if one were to wear the Deathly Hallows invisibility cloak while attempting to handle a Boggart?

Would the Boggart know you were there (like a Dementor) and react accordingly, or, could you see the true form of a Boggart while hiding under the nearly perfect and flawless cloak?

related: Could Mad-Eye Moody See the Natural State of a Boggart With His Magical Eye?

Answer 1

Probably not, but it's not certain.

What Lupin says about the Boggart somewhat implies that Boggarts decide what to turn into by seeing the person. He says that a Boggart sitting in the darkness in the cupboard wouldn't know what to turn into yet.

“So the Boggart sitting in the darkness within has not yet assumed a form. He does not yet know what will frighten the person on the other side of the door. Nobody knows what a Boggart looks like when he is alone, but when I let him out, he will immediately become whatever each of us most fears.” _{^{- Harry Potter and the Prisoner of Azkaban, Chapter 7 (The Boggart in the Wardrobe)}}

However, that could also be because there's a physical barrier. It isn't clear exactly how a Boggart sees a person. We do know that it doesn't equally sense everyone in Lupin's classroom, since Lupin moves each student forward and in front of the Boggart for them to take turns practicing.

The Boggart may still have other methods than sight for detecting people, though, and simply focused on the closest person since there were so many to choose from in the room. If it was alone with someone under an Invisibility Cloak, it may still be able to find them if a Boggart has another way of sensing the presence of a person.

Every time Lupin talks about the Boggart, he says when it sees the person.

“The Boggart will turn into a Dementor when he sees you, so we’ll be able to practise on him. I can store him in my office when we’re not using him; there’s a cupboard under my desk he’ll like.” _{^{- Harry Potter and the Prisoner of Azkaban, Chapter 12 (The Patronus)}}

This implies it does work mainly by sight.

“Or did you realise that the Boggart changed into the moon when it saw me?” _{^{- Harry Potter and the Prisoner of Azkaban, Chapter 17 (Cat, Rat and Dog)}}

Answer 2

Probably it can't see, but otherwise notice the wearer.

Harry's cloak is described by Dumbledore as the only "perfect" invisibility cloak, so I would by default assume that nothing can see through (Moody's eye is something special).

But cats can apparently "feel" the presence of someone in the invisibility cloak, either by smelling or some other way, as Filch's cat demonstrates by looking at the invisible Harry in book one (I think the chapter is called "duel at midnight").

Since a boggart is a creature that always needs to know if it's being looked at, it probably doesn't only rely on sight to notice people. And since it automatically knows everyone's biggest fear, it has telepathic abilities that probably allow it to notice when someone is observing it.