A proof of the Theta Operator Conjecture

In the context of the (generalized) Delta Conjecture and its compositional form, DAdderio, Iraci, and Wyngaerd recently stated a conjecture relating two symmetric function operators, $D_k$ and $Theta_k$. We prove this Theta Operator Conjecture, finding it as a consequence of the five-term relation of Mellit and Garsia. As a result, we find surprising ways of writing the $D_k$ operators.