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In this paper we disprove a twenty-two year old theorem about the structureof the Kauffman bracket skein module of the connected sum of two handlebodies.We achieve this by analysing handle slidings on compressing discs in ahandlebody. We find more relations than previously predicted for the Kauffmanbracket skein module of the connected sum of handlebodies, when one of them isnot a solid torus. Additionally, we speculate on the structure of the Kauffmanbracket skein module of the connected sum of two solid tori.

Kauffman Bracket Skein Module of the Connected Sum of Handlebodies: A Counterexample