I can't help but obsess over this, so I wrote down some thoughts to hopefully give you ideas. In all this, try to answer the only relevant question you have as far as the evidence: How did the perch break? Here are different scenarios to consider...
Bird Was Stolen to be Sold
1. The culprit must have been in the same building or an adjacent building who knew exactly how to get in your unit, that you left the apartment building and that you are the couple with the bird. I would suspect neighbors, land lords, janitors, managers. Their motive is to sell the bird.
2. If they are in the apartment building, so should your bird. If that's the case, he may call out to you so keep trying. Unless...
3. It is so well planned that they had someone waiting to take the bird away from the building. Which means...
4. People must have witnessed the exchange. The bird must have made some noise. The best thing to do is...
5. Ask around. Your neighbors, the landlord, people in the next building.
Bird Was Taken in Anger
1. The culprit must have been in the same building or an adjacent building who knew exactly how to get in your unit, that you left the apartment building and that you are the couple with the bird. I would suspect neighbors, land lords, janitors, managers. Their motive is driven by anger perhaps due to the noise level of the bird.
2. It would not make sense for this person to keep a bird if they found it noisy. I would not suspect the bird to be in the same building in this scenario.
3. I would still ask around the neigborhood. Someone must have withessed the bird being taken away.
4. If the bird was let loose, widen the search area. Put out flyers, contact the proper institutions.
5. I'm sorry if this may seem inconsiderate of me, but I am sure you have thought of the worst things. Its possible that an angry neighbor would want to dispose of the bird right away so he would not get caught. There are crazy people out there who would not think twice about hurting an animal.
Bird Flew Out of the Cage, Still in Apartment
1. He could be afraid and hiding somewhere, especially if he is injured.
2. I know you've looked everywhere. Just to add, check vents, behind stoves, refrigerators, washer/driers, air shafts, clothes piles, shoes.
3. To lure him out, leave his favorite treats and water in places you think would be easy for him to access.
4. Try to close toilet bowls and don't leave cooking food unattended in the off chance that he comes out and ends up in it.
5. Go on your computer or phone and play conure contact calls, in and around your unit in hopes of luring him out.
I will think of more if I can. I really hope you find your bird, or at the very least, get some sort of resolution / closure on this. Please do not give up!!! I am hoping for the best for you and your family!
~Boomer's Dad