Borel CSP complexity dichotomy conjecture

Let $\mathcal{D}$ be a finite relational structure, and let $\mathrm{CSP}_B(\mathcal{D})$ denote the set of codes for Borel structures admitting Borel homomorphisms into $\mathcal{D}$. The projective pointclasses $\mathbf{\Pi}^{1}_{1}$ and $\mathbf{\Sigma}^{1}_{2}$ classify the complexity of such sets.

Borel CSP dichotomy conjecture. For every $\mathcal{D}$, the set $\mathrm{CSP}_B(\mathcal{D})$ is either in $\mathbf{\Pi}^{1}_{1}$ or is $\mathbf{\Sigma}^{1}_{2}$-complete.

The source says that this dichotomy follows assuming $\mathbf{\Sigma}^{1}_{2}$-determinacy, but whether it holds in ZFC is open. It is intended as a descriptive-set-theoretic analogue of the finite CSP dichotomy.