My personal guess is that single-junction perovskite cells will not ever overtake single-junction silicon cells for rooftop or utility scale solar. Single junction perovskite cells may be used in applications where light weight and flexibility are advantageous, like charging portable electronics, if they can be stabilized.
Perovskite cells may compete in rooftop/utility solar with conventional silicon when incorporated into tandem cell designs -- either perovskite on silicon or a stack of different perovskites with different band gaps. That gives them the potential to exceed conventional crystalline silicon module efficiency rather than merely play catch-up. The company that seems to be furthest along with this approach is Oxford PV, which is pursuing a perovskite/silicon tandem design: