The uniform Martin's conjecture for many-one degrees

We study functions from reals to reals which are uniformly degree-invariant from Turing-equivalence to many-one equivalence, and compare them "on a cone." We prove that they are in one-to-one correspondence with the Wadge degrees, which can be viewed as a refinement of the uniform Martin's conjecture for uniformly invariant functions from Turing- to Turing-equivalence. Our proof works in the general case of many-one degrees on $\mathcal{Q}^ω$ and Wadge degrees of functions $ω^ω\to\mathcal{Q}$ for any better quasi ordering $\mathcal{Q}$.

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