Something like this.... But with 5,5 teeth of course.
Keep the green cone. Make an adapter surface (light blue), and a 44-sided cone (light pink and yellow)
(Here I inlined 48-sided primitives, but you need to make it "44-sided", matching the 5,5 teeth.)
Make a 1/8 subfile that will attach to the pink 4-4ndis and three vertices in the green cone.