I have seen this question on reddit (r/math's chatroom - to be specific). No one has answered it yet, so I thought I would pose the same question here.
The only discovery I made that is worth mentioning is this:
If it is possible to divide any triangle into convex pentagons then it is possible to divide the square as well (But not the other way around, so the triangle may as well be a waste of time).
Note: Just as I was writing this, an answer to the question was found (kind of?). But the answer is not very direct and there is still some confusion. Plus, I think this will pick the interest of people here as well. So I am going to ask it anyway.
Edit - Seems like that answer was actually wrong.