Explicit power-sum formula for higher diagonal harmonics

Let $\mathscr{H}_n^{(r)}(\mathbf{w};1,1,1)$ denote the specialization of the graded Frobenius characteristic of the higher diagonal harmonics at $q_1=q_2=1$. For a partition $\lambda\vdash n$, let $\ell(\lambda)$ be its number of parts, $z_\lambda$ the standard symmetric-group factor, $p_\lambda(\mathbf{w})$ the power-sum symmetric function, and let $\prod_{k\in\lambda}$ denote the product over the parts of $\lambda$. Explicit Frobenius formula conjecture. The specialized Frobenius characteristic is

$$\mathscr{H}_n^{(r)}(\mathbf{w};1,1,1)=\sum_{\lambda\vdash n}(-1)^{n-\ell(\lambda)}(rn+1)^{\ell(\lambda)-2}\left(\prod_{k\in\lambda}\binom{(r+1)k}{k}\right)\frac{1}{z_\lambda}p_\lambda(\mathbf{w}).$$

This gives an explicit power-sum expansion for the specialization that earlier results identify with a weighted enumeration of $r$-Dyck paths and parking functions; the formula is motivated by calculations and is stated as an additional conjecture.