Theta classes: generalized topological recursion, integrability and $\mathcal{W}$-constraints

We study the intersection theory of the $Θ^{r,s}$-classes, where $r \geq 2$ and $1 \le s \le r-1$, which are cohomological field theories obtained as the top degrees of Chiodo classes. We show that the recently introduced generalized topological recursion on the $(r,s)$ spectral curves computes the