Part 1 of Martin's conjecture for Turing invariant functions

Let $2^\omega$ be Cantor space. A function $f\colon 2^\omega\to 2^\omega$ is Turing invariant if

$$x\equiv_T y\implies f(x)\equiv_T f(y).$$

A cone of Turing degrees is a set of the form $\{x\in 2^\omega\mid x\ge_T y\}$ for some real $y$; saying that a property holds for all sufficiently large $x$ means that it holds for all $x$ in some cone.

Part 1 of Martin's conjecture. Assuming $\mathsf{ZF}+\mathsf{AD}$, if $f\colon 2^\omega\to 2^\omega$ is a Turing invariant function, then either

$$f(x)\ge_T x$$

for all $x$ in some cone, or there is some $y$ such that

$$f(x)\equiv_T y$$

for all $x$ in some cone.

Martin's conjecture aims to classify the limit behavior of functions on the Turing degrees under strong set-theoretic hypotheses. The informal formulation says that such a function is either eventually above the identity or eventually constant on a cone. The first candidate's assertion without the determinacy hypothesis is false in $\mathsf{ZFC}$; the formal statement is conjectured under $\mathsf{ZF}+\mathsf{AD}$, and its resolution is not established by the supplied text.