TIP: Before treating the upper tube edge with medium (CA) Super Glue, check to be sure there are no lifted inside spirals. If a lift is starting, glue it down now with wood glue. You won't be able to glue it down after the tube end is treated with CA glue.
Squeeze a drop of CA onto some scrap cardboard. I use a Q-tip to apply the glue to the inside edge of the tube. Your fingers can act as a depth guide while you go around the tube. If done correctly you'll have a very even line.
TIP: Don't apply CA glue to the bottom end of the tube. The low centering ring is even with the end of the tube. If the bottom of the tube were treated with CA, the wood glue on the centering ring wouldn't adhere to the sealed tube edge.
Back to the top -
Also apply some CA over the top edge. After it dries a light sanding with 400 grit on a block will square up and sharpen the edge.
The dried CA glue inside the tube top is also smoothed with 400 grit. Hold the sandpaper as shown using your fingers on the top again as a depth gauge. Don't sand lower than the CA glue coat.