D'Adderio–Iraci–Vanden Wyngaerd conjecture for the supersymmetric double coinvariant ring

Let $TDR_n$ be the quotient of

$$\mathbb{C}[x_1,\ldots,x_n,y_1,\ldots,y_n]\otimes\wedge\{\theta_1,\ldots,\theta_n,\xi_1,\ldots,\xi_n\}$$

by the ideal generated by positive-degree $\mathfrak{S}_n$-invariants. It is quadruply graded, with bosonic degrees in the $x$- and $y$-variables and fermionic degrees in the $\theta$- and $\xi$-variables. Let $(TDR_n)_{i,j}$ denote its component of homogeneous $\theta$-degree $i$ and $\xi$-degree $j$, and let $\Theta_i$ be the Theta operator indexed by the elementary symmetric function $e_i$; write $\nabla$ for the Bergeron–Garsia nabla operator.

D'Adderio–Iraci–Vanden Wyngaerd conjecture. The space $(TDR_n)_{i,j}$ vanishes whenever $i+j\geq n$. If $i+j<n$, then

$$\operatorname{grFrob}((TDR_n)_{i,j};q,t)=\Theta_i\Theta_j\nabla e_{n-i-j},$$

where $q$ tracks the $x$-degree and $t$ tracks the $y$-degree.

This conjecture gives the quadruply graded symmetric-group structure of the supersymmetric double of the diagonal coinvariant ring in terms of Theta operators and the nabla operator. The supplied text gives no evidence of a resolution status.