The conjecture that every sufficiently large integer is polite

For positive integers $a,b$, let $r(a,b)$ denote the least integer such that every red-blue coloring of $E(K_{r(a,b)})$ contains a red $K_a$ or a blue $K_b$. A positive integer $k$ is polite if

$$r(k,\lceil k/2\rceil)\leq 2^{-31}r(k,k)$$

and

$$\frac{r(k,k)-1}{r(k,k-1)}\geq 1+25\left(\frac{r(k,\lceil k/2\rceil)}{r(k,k)}\right)^{1/4}.$$

Politeness conjecture. Every sufficiently large integer is polite. The paper explains that this would follow from natural, currently unproved growth assumptions for Ramsey numbers, so it remains open.