Does every computable instance of the Laver Partition Theorem have a solution of low arithmetical complexity, or is there a computable instance requiring hyperarithmetic strength?

The Laver Partition Theorem guarantees the existence of highly structured homogeneous subtrees when an infinite tree-like object is partitioned into finitely many classes. The question is whether, when the partition function itself is computable, one can always find a homogeneous subtree whose description lands within a manageable arithmetical complexity class, or whether some computable partitions inherently force solutions that are hyperarithmetically complex or even require the full strength of Pi-1-1 comprehension to produce. This is a question about the reverse mathematical and computability-theoretic calibration of the Laver Partition Theorem relative to other Ramsey-type results on trees.