More minimal non-$\sigma$-scattered linear orders

Assuming an instance of the Brodsky-Rinot proxy principle holding at a regular uncountable cardinal $\kappa$, we construct $2^\kappa$-many pairwise non-embeddable minimal non-$\sigma$-scattered linear orders of size $\kappa$. In particular, in G\"odel's constructible universe $L$, these linear orders exist for any regular uncountable cardinal $\kappa$ that is not weakly compact. This extends a recent result of Cummings, Eisworth and Moore that takes care of all the successor cardinals of $L$. At the level of $\aleph_1$, their work answered an old question of Baumgartner by constructing from $\diamondsuit$ a minimal Aronszajn line that is not Souslin. Our use of the proxy principle yields the same conclusion from a weaker assumption which holds for instance in the generic extension after adding a single Cohen real to a model of $CH$.