Cycle Ramsey-number conjecture for loose cycles in uniform hypergraphs

Let $\mathcal{C}^k_n$ denote the $k$-uniform loose cycle with $n$ edges, and let $R(\mathcal{H},\mathcal{G})$ be the Ramsey number for hypergraphs $\mathcal{H}$ and $\mathcal{G}$. Let $k\geq 3$ be an integer. For every $n\geq m\geq 3$, the loose-cycle Ramsey-number conjecture.

$$R(\mathcal{C}^k_n,\mathcal{C}^k_m)=(k-1)n+\left\lfloor\frac{m-1}{2}\right\rfloor.$$

This is the cycle-only reduction of the preceding conjecture. The paper states that, together with the connection result relating the cycle Ramsey number to the path-related Ramsey numbers, it is sufficient to establish the full conjectured formula.