Thank you, Ben.
I’ve updated Local, and I’ve found that I can create a new install and connect to it in the browser. However, when I use Local to connect to WP Engine and pull down a site that I need to work on, I’m back to the ERR_CONNECTION_RESET. There is no information in either the local-lightning.log file or the local router error.log. It seems that as far as Local knows, everything is fine.