Again, you can certainly try to board - nothing stopping you from trying. Just make sure you have a plan B if you're refused and need to find another way home.
I agree with this observation by scylla.
I fly with Air Canada... is there any chance they will let me board?
I do not know what chance you have. I disagree with the assertion the rule is so hard and fast that it is a serious violation if the airline allows you to board. Under the eTA rules the airlines has the FINAL say and they can indeed decide to allow a PR like you to board the flight. HOWEVER, that falls way, way short of suggesting there is much chance they will.
And as far as I understand, the airline does NOT pay a penalty or suffer any other detriment if you pass through the Canadian PoE OK . . . which it appears should be NO problem.
If this ticket was issued as part of a round-trip itinerary, it is possible that your boarding pass was already approved, or that this will otherwise suffice to tip the scales toward allowing you to board. Again, this is merely what is POSSIBLE. And I have no reason to doubt the assertion by others that it is NOT much likely . . . other than, again, there is the fact that the decision to allow boarding is ultimately for the airlines to make, which seems to be overlooked by some.
However this goes, please let us know. Similar situations are actually fairly common. Even if you benefit from an exception, and the airline allows you to board the flight, that does NOT mean anyone else should expect that to happen for them . . . but if you let us know how it actually goes, that will help us have a better idea if there is much chance at all that an airline might allow for such an exception.
Good luck!