Haxell–Łuczak–Peng–Rödl–Ruciński–Skokan conjecture for tight-cycle Ramsey numbers

For $k e 3$, let $C_{kn+i}^{(k)}$ denote the $k$-uniform tight cycle on $kn+i$ vertices, and let $r(H)$ be the two-colour Ramsey number of a $k$-graph $H$. For $0 c i c k-1$, set

$$d \coloneqq \gcd(k,i).$$

Haxell–Łuczak–Peng–Rödl–Ruciński–Skokan conjecture. We have

$$r(C_{kn+i}^{(k)})=(1+o(1))\frac{d+1}{d}kn.$$

The paper proves the case $i=0$ for every $k\geq3$, extending previously known cases; the remaining values of $i$ are left open.