Martin's conjecture for T-invariant functions on the Turing degrees

Let $AD$ denote the axiom of determinacy. A function $f\colon\mathcal{P}(\mathbb{N})\to\mathcal{P}(\mathbb{N})$ is $T$-invariant if $A\equiv_T B$ implies $f(A)\equiv_T f(B)$. A function is increasing on a cone if $\operatorname{Id}\leq_T^{\triangledown} f$, and constant on a cone if its values are Turing equivalent on some cone.

Martin's conjecture. Assume $AD$. Then:

1. If $f\colon\mathcal{P}(\mathbb{N})\to\mathcal{P}(\mathbb{N})$ is $T$-invariant, then $f$ is either increasing on a cone or constant on a cone.
2. The increasing $T$-invariant functions in $\mathcal{P}(\mathbb{N})$ are well-ordered up to Turing equivalence on a cone, and the successor in this well-order is given by the Turing jump.

Both parts remain open, although they have been proved for particular classes of functions, including uniformly $T$-invariant functions.