Apparently, I still do not know how to work this system. I thought I replied to you. At first, I decided to go with thread. Then I discovered it was a challenge to find it. I had bought the cork (the right thickness) and cut it to size placing it on the tenon. I then connected the pieces without gluing to see the fit.
Everything fit tight and am now playing without risking glue yet. That’s the update.
I hope you get this message.
Thanks for all your perseverance and help, Lou