The odds of making the connection are very high, even without running. I’m curious, though, what you expect informing the flight crew would accomplish? There’s certainly no harm, but it’s not going to do anything. The gate agent for the PHX-LIH flight will be aware of the status of connecting passengers and will decide, based primarily on AA policy, when to close the door. But honestly this does look like the best and a reasonably safe. I’d prefer a longer connection, but also would expect to make this.