The maximum odd-cycle and path likelihood conjecture

Let $m\geq 3$, and let $\mu$ be an edge probability mass. For a graph $H$, write $\beta(\mu;H)$ for the total $\mu$-weight of copies of $H$, and let $C_m$ and $P_{m+1}$ denote the cycle on $m$ vertices and the path on $m+1$ vertices, respectively.

Maximum likelihood conjecture. For every $m\geq 3$ and every edge probability mass $\mu$,

$$2m\cdot\beta(\mu;C_m)+\beta(\mu;P_{m+1})\leq {2\over m^{m-1}},$$

with equality if and only if $\mu$ is the uniform distribution on $E(C_m)$.

The paper establishes a bound with constant $2.6947$ for $m\geq 5$, leaving the sharper constant $2$ and its equality characterization as the main question. The conjecture is therefore open.