Ramsey numbers of uniform loose paths and cycles

Let $\mathcal{P}^k_n$ and $\mathcal{C}^k_n$ denote the $k$-uniform loose path and loose cycle with $n$ edges, respectively, and let $R(H_1,H_2)$ be the two-colour Ramsey number for hypergraphs $H_1$ and $H_2$. The loose path-and-cycle Ramsey conjecture. Let $k\geq 3$ be an integer. For every $n\geq m\geq 3$,

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

This extends the completely determined $3$-uniform case to all uniformities $k\geq 3$; the source presents it as a conjecture, and no resolution is supplied here.