Pokrovskiy–Sudakov cycle Ramsey goodness conjecture

Let $R(G,H)$ denote the two-colour Ramsey number, let $C_n$ be the cycle on $n$ vertices, and let $\chi(H)$ and $\sigma(H)$ denote the chromatic number and minimum colour-class size of $H$, respectively. Pokrovskiy–Sudakov conjecture. There exists $C>0$ such that if $H$ is a graph and $n\geq C|H|$, then

$$R(C_n,H)=(\chi(H)-1)(n-1)+\sigma(H).$$

The source presents this as open and notes its relation to a conjecture of Allen, Brightwell and Skokan.