The computable Scott sentence conjecture

A computable structure is a structure whose atomic diagram is computable, and a Scott sentence is a sentence characterizing the structure up to isomorphism among countable structures. The classes $\Pi_n$ and $\Sigma_{2n}$ refer to the corresponding infinitary quantifier-complexity levels.

Computable Scott sentence conjecture. For each even $n\geq 2$, there is a computable structure $\mathcal{M}$ with a $\Pi_n$ Scott sentence but no computable $\Sigma_{2n}$ Scott sentence.

This conjecture from Scott analysis is presented as an implication of the first back-and-forth complexity assertion. The source gives no resolution, so the conjecture remains open.