Remarks on Halin's end-degree Conjecture

We prove new instances of \emph{Halin's end degree conjecture} (\emph{HC}) within $\mathrm{ZFC}$. In particular, we prove that there is a proper class of cardinals $\kappa$ for which Halin'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 \emph{HC} and the \emph{Singular Cardinal Hypothesis}, deriving consistency strength from failures of the former. We verify that Halin's conjecture fails on finite intervals of successors of singular cardinals in Meremovich's model, this is a new independence result on \emph{HC}.