Interesting pipe.
A long, gradual taper to the mortise, and a perfectly matching taper to the tenon (the stem), should provide an adequate friction fit, without using wax.
If the mortise is already bored to a diameter that nearly matches the diameter of the stem, then that approach will not work. You might create a suitably tapered adapter of wood, plastic, horn or metal, that can be countersunk and glued into the wood of the existing pipe.
A cruder method is to coat the exterior of the stem tenon with plastic epoxy--to increase its diameter, then after it is cured, sand it to fit.
Whatever method you choose, you will need to be able to remove the stem frequently for cleaning.
Bob