Analytic spread formula for pairs of cycles

Let $C_n$ and $C_m$ be cycles with $n,m\geq 3$, let $J_{C_n,C_m}$ be their associated binomial edge ideal, and let $\ell(J_{C_n,C_m})$ denote its analytic spread. Assume that the coefficient field $F$ has characteristic zero. Cycle analytic-spread conjecture. Then

$$\ell(J_{C_n,C_m})=\begin{cases} nm & \text{if $n$ and $m$ are odd},\\\\ nm-1 & \text{if exactly one of $n$ or $m$ is even},\\\\ nm-2 & \text{if $n$ and $m$ are even}. \end{cases}$$

In particular, $(C_n,C_m)$ is a basis if and only if $n$ and $m$ are odd. The formula is supported by computations using random matrices in Macaulay2, and the supplied text gives no proof or resolution.