The analytic torsion formula for Shannon products with linear graphs

Let $C_n$ be a circle and $L_n$ a linear graph with $n$ vertices, with $C_n * L_n$ their Shannon product. The linear-graph product conjecture.

$$A(C_n * L_n) = n^2 m/(3m-2).$$

The notation is inconsistent: the left-hand side uses $L_n$, while the right-hand side contains $m$. The supplied text gives no supporting discussion or proof.