The off-diagonal Erdős–Rado bound for monochromatic and lexical $K_4$

Let $\operatorname{ER}(4,4,r)$ denote the off-diagonal unordered Erdős–Rado number when monochromatic and lexical copies of $K_4$ are forbidden and a rainbow clique has order $r$. The $K_4$ off-diagonal Erdős–Rado conjecture. There exists a constant $c>0$ such that, for every $r\geq3$,

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

This is identified as the smallest case in which the paper does not obtain the correct asymptotic, and is presented as a potentially easiest case of the general problem.