Every subset of $\mathbb N \times \mathbb N$ can be viewed as a relation on $\mathbb N$. The set $\mathcal P(\mathbb N \times \mathbb N)$ of all relations on $\mathbb N$ has a natural topology with which it is homeomorphic to the Cantor space. A pretty well-known fact from descriptive set theory is:

The subset $\mathsf{WO}$ of well orderings of $\mathbb N$ is a coanalytic subset of $\mathcal P(\mathbb N \times \mathbb N)$.

This is one of the most natural examples out there of a subset of a Polish space that is coanalytic but not Borel. This coanalytic set stratifies nicely into an increasing union of $\omega_1$ Borel sets:

For each $\alpha < \omega_1$, the set $\mathsf{WO}_{\alpha}$ of all well orderings of $\mathbb N$ with order type $\alpha$ is Borel.

**Question:** Given $\alpha < \omega_1$, is there a $G_\delta$ set $A \subseteq \mathcal P(\mathbb N \times \mathbb N)$ such that $\mathsf{WO}_{\alpha} \subseteq A \subseteq \mathsf{WO}$?

*Motivation:*
If the answer is positive, then we would be able to write this coanalytic set as a union $\bigcup_{\alpha < \omega_1} A_\alpha$ of $G_\delta$ sets, which would imply that there is a partition $\{ A_\alpha \setminus \bigcup_{\xi < \alpha} A_\xi :\, \alpha < \omega_1 \}$ of $\mathsf{WO}$ into $F_{\sigma \delta}$ sets. Because $\mathsf{WO}$ is an example of a complete coanalytic set, this would then imply that every coanalytic set can be partitioned into $\aleph_1$ $F_{\sigma \delta}$ sets, answering another recent question of mine.