On Halin's end-degree Conjecture and $\omega$-strong cardinals

We prove new instances of \emph{Halin's end degree conjecture} within $\mathrm{ZFC}$. In particular, we prove that there is a proper class of cardinals $\kappa$ for which Haliln's end-degree conjecture holds. This answers two questions posed by Geschke, Kurkofka, Melcher and Pitz in 2023. Furthermore, we comment on the relationship between Halin's conjecture and the \emph{Singular Cardinal Hypothesis}, deriving consistency strength from failures of the former. We also show that Halin's conjecture fails on finite intervals of successors of singular cardinals in Meremovich's model.
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