Does a $\kappa$-Kurepa tree imply we have a slim $\kappa$-Kurepa tree? Take a $\kappa$-Kurepa tree to be a tree with more than $\kappa$ branches, of height $\kappa$, each level having cardinality less than $\kappa$ A slim $\kappa$-Kurepa tree is the same but the cardinality of each level $\alpha<\kappa$ has cardinality $\leq|\alpha|$. Is it true in general the existence of a $\kappa$-Kurepa tree implies a slim one exists?

Well, if $\kappa$ is a successor, then any $\kappa$-Kurepa tree is slim (or, at least, is slim above some level).

But the implication is not true in general. For example, if $\kappa$ is a strong limit, then the full binary tree of height $\kappa$ is a $\kappa$-Kurepa tree. But if $\kappa$ additionally has some compactness properties (for example, if it is measurable) then there cannot be any slim $\kappa$-Kurepa trees.