Conjecture on the complexity of definable orbits of c.e. sets

Let $A$ be a c.e. set whose orbit is properly $\Delta^0_{\alpha}$ for a finite $\alpha>8$, as in Theorem 3.2, and let $\mathcal{L}_{\omega_1,\omega}$ denote the infinitary logic used to describe orbits in the lattice $\mathcal{E}$ of computably enumerable sets. Orbit-complexity conjecture. The above orbits can be built so that the complexity of the $\mathcal{L}_{\omega_1,\omega}$ formula describing the orbit is close to $\alpha$. This would refine the known existence of properly $\Delta^0_{\alpha}$ orbits by relating their arithmetical complexity to the complexity of infinitary formulas defining them; the source gives no resolution of the conjecture.