Okay, this gives me a better picture.
You say that he bites her when you are sure you are not looking at him. Well, when he bites her, you certainly turn and look at her, right? Oh boy, he's got your attention! And he's got your wife's attention! Even though the screaming and crying might be a negative response, it is still a response and that is what your bird is seeking.
Birds often don't associate "Oh, I did this bad thing, now they are going to wrap me in a scarf and put me back on the cage, maybe I shouldn't bite again." The cause-effect scenario is more immediate with them. It's more like "I bit her and now I have everyone's attention! Yay! Wait, what are you doing? Why are you putting me back? Hey, this isn't fair, I what did I do?" Total disconnect. Which is why positive punishment does not work.
Positive punishment is when you introduce something to the bird that the bird does not like. Such as getting wrapped in a scarf.
You are also using negative punishment when you put him back in the cage. With negative punishment, you are taking something away that he finds pleasurable, which in this case is your attention and out-of-cage time.
Now, negative punishment can be useful by removing both you and your wife's attention
immediately after the behavior. Your wife needs to do her best NOT to give a response the next time your conure bites her. She shouldn't say anything. She should just quietly put him down (on his play stand, on the table, whatever is the CLOSEST SAFE SURFACE to wherever she is) and turn his back to him for a few minutes. You should act as though nothing has happened and continue doing what you are doing. Do NOT look at him or give him attention for the biting behavior.
There is more that you can do beyond this point to make it a lasting behavior, but this would be the first step.
That being said, I am not an expert. This is just my interpretation of the situation. If you DO want to learn more from an expert, I recommend you check out Barbara Heidenreich's webinar (#3, on biting and aggressive behavior). I have not watched it (yet), but I have seen her other webinars she is considered the best in the field. She uses positive reinforcement (you did a good thing, here is a treat!) in order to shape behavior.
I hope this helps! Please keep us posted, and if you have any other questions or concerns, please ask away!
Edit: I almost forgot the link for the webinars!!! Here it is:
http://www.goodbirdinc.com/parrot-store-digital.htm
Edit (again): I also forgot to mention that webinar #8 is free - it is more general and talks specifically about rescue birds, but it is still helpful