Ramsey numbers of $4$-uniform loose cycles

Gy'arf'as, S'ark"ozy and Szemer'edi proved that the $2$-color Ramsey number $R({\mathcal{C}}_n^k,{\mathcal{C}}_n^k)$ of a $k$-uniform loose cycle ${\mathcal{C}}_n^k$ is asymptotically $\frac{1}{2}(2k-1)n,$ generating the same result for $k=3$ due to Haxell et al. Concerning their results, it is conjectured that for every $n\ge m\ge 3$ and $k\ge 3,$ $$R({\mathcal{C}}_n^k,{\mathcal{C}}_m^k)=(k-1)n+\lfloor \frac{m-1}{2}\rfloor \mathrm{.}$$ In $2014$, the case $k=3$ is proved by the authors. Recently, the authors showed that this conjecture is true for $n=m\ge 2$ and $k\ge 8$. In this paper, we show that the conjecture holds for $k=4$ when $n>m$ or $n=m$ is odd. When $n=m$ is even, we show that $R({\mathcal{C}}_n^4,{\mathcal{C}}_n^4)$ is between two values with difference one.