Are you absolutely certain that your hive is queenless? Three weeks post swarm can be a short time to get a new queen emerged, matured, mated and then laying long enough to see her new brood readily. They're usually pretty good at getting a replacement. Having said that, they do sometimes fail.
By all means, put in a frame with eggs. What do you have to lose other than a few days? If they start making queen cells you'll know for sure. Then you can decide what to do. I don't know how it is in MN but here I still have plenty of drones. The real question, in my opinion, is if they're queenless are there enough bees or time to make a cluster that'll survive a MN winter?
I've heard it said that you want to take your losses in the Fall. I think its good advice.
Let me never fall into the vulgar mistake of dreaming that I am persecuted whenever I am contradicted. - Emerson