Martin's conjecture on degree-invariant functions

Assume $\mathsf{ZF}+\mathsf{DC}+\mathsf{AD}$. A function $f:2^\omega\to2^\omega$ is degree-invariant if $A\equiv_T B$ implies $f(A)\equiv_T f(B)$, and a property holds almost everywhere (a.e.) with respect to Martin Measure when it holds on a cone. Write $f'$ for the function defined by $f'(x)=f(x)'$, and let $\leq_m$ denote the ordering used for the ranks of degree-invariant functions.

Martin's conjecture. Under these assumptions: (I) if $f$ is degree-invariant and is not increasing a.e., then $f$ is constant a.e.; and (II) $\leq_m$ pre-well-orders the set of degree-invariant functions that are increasing a.e., with $f'$ having $\leq_m$-rank $\alpha+1$ whenever $f$ has $\leq_m$-rank $\alpha$.

Martin's conjecture classifies degree-invariant functions by their almost-everywhere behavior, analogous to the classification of natural objects by iterates of the Turing jump. The source states that it remains an open problem, and notes that under suitable large-cardinal hypotheses it can be viewed as a conjecture about functions in $L(\mathbb{R})$.