Uniformization conjecture for Borel homomorphisms from arithmetic to Turing equivalence

Let $\equiv_A$ denote arithmetic equivalence and $\equiv_T$ Turing equivalence on $2^\omega$. A homomorphism from $\equiv_A$ to $\equiv_T$ preserves equivalence, and uniformity means that there is a function $u:\omega^2\to\omega^2$ witnessing, for every arithmetic reduction pair $(i,j)$, a corresponding Turing reduction pair. Uniformization conjecture. If $f$ is a Borel homomorphism from $\equiv_A$ to $\equiv_T$, then there exists a Borel homomorphism $g$ from $\equiv_A$ to $\equiv_T$ such that $f(x)\equiv_Ag(x)$ on an arithmetic cone and $g$ is uniform: for all $x,y$, if $x\equiv_Ay$ via $(i,j)$, then $g(x)\equiv_Tg(y)$ via $u(i,j)$. The question concerns whether nonuniform Borel homomorphisms admit uniform representatives on a cone; the source gives no resolution.