The odd-order dense quasirandom edge-colouring conjecture

For $0<p<1$, suppose there exist $\varepsilon,\eta>0$ and $n_0\in\mathbb{N}$ such that $G$ is a lower-$(p,\varepsilon)$-regular graph on $n\geq n_0$ vertices, $n$ is odd, and $\Delta(G)-\delta(G)\leq\eta n$. Odd-order quasirandom edge-colouring conjecture. Under these hypotheses,

$$\chi'(G)=\Delta(G)\quad\text{if and only if}\quad \sum_{x\in V(G)}\bigl(\Delta(G)-d_G(x)\bigr)\geq\Delta(G).$$

This is presented as an interesting possible analogue of the even-order quasirandom result; the source does not claim it as proved.