The analytic torsion formula for Shannon products with complete graphs

Let $C_n$ be a circle and $K_m$ a complete graph, with $C_n * K_m$ their Shannon product. The complete-graph product conjecture.

$$A(C_n * K_m) = n^2/m.$$

This is presented as an experimentally observed formula for homotopy cylinders; the paper does not provide a proof.