Multiple-copy Ramsey numbers for loose and tight hypergraph paths and cycles

Let $R(\mathcal{F},\mathcal{G})$ denote the Ramsey number for hypergraphs, and let $m\mathcal{F}$ denote the disjoint union of $m$ copies of $\mathcal{F}$. For $k$-uniform hypergraphs, write $\mathcal{P}_r^k$, $\mathcal{C}_r^k$, and $\hat{\mathcal{P}}_r^k$ for the relevant path and cycle hypergraphs.

Multiple-copy path and cycle conjecture. For every $k\geq 3$, $m\geq n$ and $r\geq s\geq 3$,

$$R(m\mathcal{P}_r^k,n\mathcal{P}_s^k)=R(m\mathcal{P}_r^k,n\mathcal{C}_s^k)=((k-1)r+1)m+\Big\lfloor\frac{s+1}{2}\Big\rfloor n-1,$$ $$R(m\hat{\mathcal{P}}_r^k,n\hat{\mathcal{P}}_s^k)=(r+k-1)m+(1+\lfloor\frac{s-1}{k}\rfloor)n-1,$$ $$R(m\mathcal{C}_r^k,n\mathcal{C}_s^k)=(k-1)rm+\Big\lfloor\frac{s+1}{2}\Big\rfloor n-1,$$

and, if $r>s$,

$$R(m\mathcal{C}_r^k,n\mathcal{P}_s^k)=(k-1)rm+\Big\lfloor\frac{s+1}{2}\Big\rfloor n-1.$$

The first and third formulas extend the established $k=3$ results and the lower bound for multiple loose cycles; determining whether the natural lower bound is always exact remains open in general, with analogous questions for loose paths, tight paths, and tight cycles.