Nikiforov's asymptotic cycle–complete graph Ramsey conjecture

Let $r(C_p,K_r)$ denote the Ramsey number for a cycle $C_p$ versus a complete graph $K_r$, where $p$, $r$, and $k$ are positive integers. Nikiforov's asymptotic Ramsey conjecture. For every $k$ there exists $r_0=r_0(k)$ such that, whenever $r>r_0$ and $p>r^{1/k}$,

$$r(C_p,K_r)=(p-1)(r-1)+1.$$

This conjecture would substantially extend the range in which the Erdős–Faudree–Rousseau–Schelp formula is known, far beyond the linear range established in the paper. The supplied text calls it more challenging and gives no resolution.