Can I hard prune my privet hedge in June?

I want to tackle a really overgrown privet hedge. I think the original hedge was two feet wide and five feet tall. It is now up to eight feet wide and twelve feet tall, overhangs the pavement by at least two feet. Total length on three sides of the garden is about 80 feet

I want to hard prune the outside first cutting it back by at least two feet. I'm planning to leave the inside for the time being (just a trim) to keep the privacy.

I know it's later than the optimum time for such a cutback, but can I do it now (late June)? I know it will take at least a year to recover. And yes I will fertilize after the pruning.