Ramsey numbers of $5$-uniform loose cycles

Gy'{a}rf'{a}s et al. determined the asymptotic value of the diagonal Ramsey number of ${\mathcal{C}}_n^k$, $R({\mathcal{C}}_n^k,{\mathcal{C}}_n^k),$ generating the same result for $k=3$ due to Haxell et al. Recently, the exact values of the Ramsey numbers of 3-uniform loose paths and cycles are completely determined. These results are motivations to conjecture that for every $n\geq m\geq 3$and$k\geq 3,$ $$R({\mathcal{C}}_n^k,{\mathcal{C}}_m^k)=(k-1)n+\lfloor \frac{m-1}{2}\rfloor ,$$ as mentioned by Omidi et al. More recently, it is shown that this conjecture is true for $n=m\ge 2$ and $k\ge 7$ and for $k=4$ when $n>m$ or $n=m$ is odd. Here we investigate this conjecture for $k=5$ and demonstrate that it holds for $k=5$ and sufficiently large $n$.