The delta-operator formula for the Frobenius image of the diagonal coinvariant quotient

Let ${\mathbf{x}}_n=(x_1,\dots,x_n)$ and ${\mathbf{y}}_n=(y_1,\dots,y_n)$ be two sets of variables, and let $DR_{n,k}$ be the bigraded $S_n$-module quotient by the ideal generated by the polarized power sums $p_{a,b}({\mathbf{x}}_n,{\bf y}_n)$ with $a+b\geq k+1$ and by the monomials $z_1\cdots z_n$ with $z_i\in\{x_i,y_i\}$. Write ${\mathrm{grFrob}}(DR_{n,k};q,t)$ for its bigraded Frobenius image, and let $\Delta_f$ denote the delta operator on symmetric functions. Delta-operator conjecture. The bigraded Frobenius image of $DR_{n,k}$ is

$${\mathrm{grFrob}}(DR_{n,k};q,t)=\Delta_{h_k e_n}e_n=\Delta_{s_{k+1,1^{n-1}}}e_n=\Delta_{h_k}\nabla e_n.$$

This conjecture connects the quotient defined by polarized power sums with the delta-operator and nabla formalisms in the theory of Macdonald polynomials. It gives a precise symmetric-function prediction for the full bigraded $S_n$-module structure.