The analytic torsion formula for Shannon products with wheel graphs

Let $C_n$ be a circle and $W_m$ the wheel graph with $m$ vertices, with $C_n * W_m$ their Shannon product. The wheel-product conjecture.

$$A(C_n * W_m) = n^2 m/(4m+1).$$

The formula is suggested by computations for $m=5,6,7$ and is not proved in the supplied text.