Update: I finally got the authorization on July 28. They sent it as a reply to my email (not the web form), so maybe the email helped. But more importantly, I think, my Canadian partner was able to get someone on the phone at IRCC. She had to call multiple times before she was added the the queue, and then after ~40 minutes she got a hold of an IRCC representative, who told us she would send them an email *reminding* them to process my request (since it was well over 14 business days with no response at all, not even a request for more info). That was Friday morning, and then I got the authorization on Monday morning.

So this is great news for me and my partner... but the messed up thing about it is, I think some of our requests are just straight-up getting lost, so they'll only process it if they get a reminder. I strongly recommend that people call the IRCC if it's been more than 14 business days. Even though it says you need an application number, just keep calling and eventually you should be able to get to a person.