Steel's cone-equivalence 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, and a function is uniformly invariant when its many-one equivalences are witnessed uniformly from the Turing-equivalence programs. Assume $\mathsf{AD}$. Steel's cone-equivalence conjecture. If $f$ is $(\equiv_T,\equiv_m)$-invariant, then there is a uniformly $(\equiv_T,\equiv_m)$-invariant function $g$ such that $f(x)\equiv_m g(x)$ on a Turing cone of $x$. This is presented as a strengthening of the preceding conjecture, replacing cone reducibility with pointwise many-one equivalence on a cone; the supplied text does not resolve it.