Tautological ring of strata of differentials

Strata of $k$-differentials on smooth curves parameterize sections of the $k$-th power of the canonical bundle with prescribed orders of zeros and poles. Define the tautological ring of the projectivized strata using the $\kappa$ and $\psi$ classes of moduli spaces of pointed smooth curves along with the tautological class $\eta$ of the Hodge bundle. We show that if there is no pole of order $k$, then the tautological ring is generated by $\eta$ only, and otherwise it is generated by the $\psi$ classes corresponding to the poles of order $k$.