The conjecture that all odd cycles are strongly common

Let $W$ be a graphon, meaning a symmetric measurable function $W:[0,1]^2\to[0,1]$, and let $t(H,W)$ denote the homomorphism density of a graph $H$ in $W$. A graph $H$ is strongly common if

$$t(H,W)+t(H,1-W)\geq t(K_2,W)^{e(H)}+t(K_2,1-W)^{e(H)}$$

for every graphon $W$. Odd-cycle strong-commonness conjecture. Every odd cycle is strongly common. Strong commonness implies commonness, and the claim is motivated by the known cases of the triangle and the $5$-cycle; the general assertion remains open.