Cholak–Epstein conjecture on outer splitting and automorphism to low-two sets

Let $A$ be a c.e. set. It has the outer splitting property if there are computable functions $f,h$ such that, for every $e$, $W_e=W_{f(e)}\sqcup W_{h(e)}$, $W_{f(e)}\cap\overline{A}$ is finite, and if $W_e\cap\overline{A}$ is infinite then $W_{f(e)}\cap\overline{A}$ is nonempty. Let $\overline{A}$ be semilow$_2$, and let a c.e. set be low$_2$ when its second Turing jump is equivalent to $\mathbf{0}^{(2)}$. Cholak–Epstein conjecture. Every set $A$ with the outer splitting property and semilow$_2$ complement is automorphic to a low$_2$ set. The conjecture would strengthen the known result that these hypotheses imply $\mathcal{L}^*(A)\approx\mathcal{L}^*(\emptyset)$; the source gives no resolution.