Asymptotic tower formula for color-avoiding Ramsey numbers of monotone paths

Fix an integer $p\geq 1$. Let $A_k(n;q,p)$ denote the color-avoiding Ramsey number for monotone paths, and let $T_h(x)$ denote a tower function of height $h$. Asymptotic tower formula. There exists a positive integer $q_0$ depending only on $p$ and a positive two-variable function $\gamma$ such that, for all $q\geq q_0$ and sufficiently large $n$,

$$A_k(n;q,p)=T_{\Theta(k/p)}\left(n^{\gamma(q,p)}\right),$$

where $\gamma(\cdot,p)$ is increasing and tends to infinity. This conjecture gives a precise form of the expected tower-height growth and strengthens the preceding qualitative prediction; the supplied text gives no evidence that it has been resolved.