He had to have spoken at some point between the Twins and King’s Landing even if we didn’t see it. That’s a long ride!
Besides, Jaime can’t be dead, he’s still got plot armor. You don’t kill a character who is only halfway through a redemption arc, that’s writing 101.
I agree, Jamie is a redemption character.. can’t say it won’t happen, but I think odds that Arya killed him are low.... I still see the most likely path that he kills his sister in the end.