The lexical-$K_4$ off-diagonal Erdős–Rado bound

For $m\geq4$, let $\operatorname{ER}(m,4,r)$ denote the off-diagonal unordered Erdős–Rado number in the setting where monochromatic cliques have order $m$, lexical cliques have order $4$, and rainbow cliques have order $r$. The lexical-$K_4$ Erdős–Rado conjecture. For every $m\geq4$, there exists a constant $c=c(m)>0$ such that, for every $r\geq3$,

$$\operatorname{ER}(m,4,r)\leq c\left(\frac{r^3}{\log r}\right)^2.$$

The paper gives an upper bound of order $r^{m+3+o(1)}$ when only a lexical $K_4$ is forbidden, and conjectures that the true exponent of $r$ is substantially smaller.