The odd-partite independent-transversal blowup conjecture

Let $\mathcal{G}_r(n)$ be the class of $r$-partite graphs with parts of size $n$. An independent transversal with exactly $s$ vertices in each part is denoted by $IT(s)$. Odd-partite blowup conjecture. For every odd integer $r\ge 5$ and every integer $s\ge 2$, there should exist constants $C>0$ and $n_0\in\mathbb{N}$ such that every graph $G\in\mathcal{G}_r(n)$ with $n\ge n_0$ and

$$\Delta(G)\le \frac{r-1}{2r-4}n-Cn^{1-1/s}$$

contains an $IT(s)$. This is proposed as an analogue of the corresponding result for $r=3$; the paper presents it as an open direction, with tightness related to the Zarankiewicz number.