Borel decomposition into an injective part and a countable-image part

Ascertain whether for every Borel function f between Polish zero-dimensional spaces there exists a Borel partition (A,B) of the domain such that the restriction f|A is continuously equivalent to an injection and the restriction f|B has countable image.

Background

This decomposition conjecture, if true, would separate complexity in Borel functions into a part reducible to injectivity and a part with countable image. It would provide a pathway to proving bqo for broader function classes by handling injective and countable-image components separately.