Calculus of constructions, type-in-type and recursion

Does adding type-in-type to the calculus of constructions lead to (general) recursion? Such that one can write the Y combinator.

• You may find this answer by Herman Geuvers to a similar question useful. And the whole thread is very illuminating. – Anton Trunov Aug 25 '19 at 17:50
• @AntonTrunov you might as well turn your comment into an answer. – cody Aug 25 '19 at 21:50
• So if I understand correctly you can not write the Y-combinator but only "looping" combinators. So you can not write (general) recursive functions in type-in-type. – Labbekak Sep 7 '19 at 8:18