D'Adderio–Iraci–Vanden Wyngaerd Theta conjecture for the $(2,2)$-bosonic-fermionic coinvariant ring

Let $R_n^{(2,2)}$ be the $(2,2)$-bosonic-fermionic coinvariant ring, and let $\operatorname{Frob}(R_n^{(2,2)};q,t;u,v)$ denote its multigraded Frobenius series. Here $\Theta_{e_k}$ and $\Theta_{e_\ell}$ are Theta operators indexed by elementary symmetric functions, $\nabla$ is the nabla operator, and $e_m$ is the elementary symmetric function. Theta conjecture. For all $n \geq 1$,

$$\operatorname{Frob}(R_n^{(2,2)}; q,t;u,v) = \sum_{ k + \ell < n} u^k v^\ell\Theta_{e_k}\Theta_{e_\ell}\nabla e_{n-k-\ell}.$$

This conjecture gives a symmetric-function formula for the multigraded Frobenius series and is used as the starting point for specializations to other bosonic-fermionic coinvariant rings. Its resolution is not specified in the source.