Finite-basis conjecture for functions beyond every countable Baire class

For a countable ordinal $\alpha<\omega_1$, consider the class of functions that are not of Baire class $\alpha$, ordered by topological embeddability. A finite basis for an upward-closed class means a finite collection of functions such that every member of the class contains one of them under topological embeddability, and every function containing one of them belongs to the class. Finite-basis conjecture. For every $\alpha<\omega_1$, the class of functions that are not Baire class $\alpha$ admits a finite basis for topological embeddability. The conjecture generalizes finite-basis results for several classes of discontinuous or non-Baire-class-1 functions, while the paper does not establish it for all countable Baire classes.