The low$_2$ supersets lattice conjecture

Let $A$ be a computably enumerable set. Write $\mathcal{E}^*$ for the lattice of all computably enumerable sets modulo finite sets, and let $L^*(A)$ denote the lattice of supersets of $A$ modulo finite sets. A computably enumerable set $A$ is low$_2$ when its second Turing jump satisfies $A”\equiv_T\emptyset”$. The low$_2$ supersets lattice conjecture. If $A$ is low$_2$, then

$$L^*(A)\cong \mathcal{E}^*.$$

Soare proved the analogous result for low computably enumerable sets, while Maass characterized effective isomorphism with $\mathcal{E}^*$ using semilow$_{1.5}$. The conjecture asks whether the conclusion extends to all low$_2$ computably enumerable sets and remains open.