Harrington's conjecture on representatives of c.e. set orbits

Let $A$ be a computably enumerable set and let $\boldsymbol{d}$ be a Turing degree. Write $A'$ for the Turing jump of $A$, and let $\mathcal{L}^*(A)$ denote the lattice structure associated with $A$ modulo finite sets. Harrington's conjecture. If

$$A' \leq_T \boldsymbol{d}',$$

then there is a computably enumerable set $\hat{A}\in\boldsymbol{d}$ such that

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

This asks whether the jump inequality is sufficient to find, within the degree $\boldsymbol{d}$, a representative whose orbit-invariant lattice structure agrees with that of $A$. The source presents it as a conjecture of Harrington; no resolution is supplied here.