Steel's conjecture for $(\equiv_T,\equiv_m)$-invariant functions

Let $2^{\omega}$ be Cantor space. A function $f\colon 2^{\omega}\to 2^{\omega}$ is $(\equiv_T,\equiv_m)$-invariant if Turing-equivalent inputs have many-one-equivalent outputs; it is uniformly $(\equiv_T,\equiv_m)$-invariant if the witnessing many-one reductions are uniformly determined by the Turing-equivalence programs. Write $f\equiv_m^\bigtriangledown g$ when $f$ and $g$ are equivalent under $\leq_m^\bigtriangledown$ on a Turing cone. Assume $\mathsf{AD}+\mathsf{DC}$. Steel's conjecture. For every $(\equiv_T,\equiv_m)$-invariant $f$, there is a uniformly $(\equiv_T,\equiv_m)$-invariant function $g$ such that $f\equiv_m^\bigtriangledown g$. This conjecture would reduce the study of all such invariant functions to the uniformly invariant ones; the supplied text says it implies the preceding Martin conjecture by Kihara–Montalbán's work, but gives no resolution.