A note on increasing paths in countable hypergraphs

An old result of M\"uller and R\"odl states that a countable graph $G$ has a subgraph whose vertices all have infinite degree if and only if for any vertex labeling of $G$ by positive integers, an infinite increasing path can be found. They asked whether an analogous equivalence holds for edge labelings, which Reiterman answered in the affirmative. Recently, Arman, Elliott, and R\"odl extended this problem to linear $k$-uniform hypergraphs $H$ and generalized the original equivalence for vertex labelings. They asked whether Reiterman's result for edge labelings can similarly be extended. We confirm this for the case where $H$ admits finitely many $\beta$-cycles involving any fixed vertex.