A model with Suslin trees but no minimal uncountable linear orders other than ${\omega }_1$ and $-{\omega }_1$

We show that the existence of a Suslin tree does not necessarily imply that there are uncountable minimal linear orders other than ${\omega }_1$ and $-{\omega }_1$, answering a question of J. Baumgartner. This is done by a Jensen-type iteration, proving that one can force CH together with a restricted form of ladder system uniformization on trees, all while preserving a rigid Suslin tree.