A-type Kostka polynomial formula

Let $\mathcal{P}$ denote the set of partitions, let $\mu\in\mathcal{P}$ and $\nu\in\mathbb{Z}^n$ satisfy $l(\mu)\leq n$ and $|\mu|=|\nu|$, and let $K_{\mu\nu}$ be the corresponding Kostka polynomial. Write $\rho$ for the Weyl vector of $A_{n-1}$, $Q$ for its root lattice, $S_n$ for the symmetric group, and $\epsilon$ for the sign character. For a composition $\eta$, let $(q)_\eta=\prod_{i\geq1}(q)_{\eta_i}$, let $m(\eta)$ be its multiplicity vector, and use the Gaussian multinomial notation displayed in the source. The A-type Kostka polynomial conjecture.

$$K_{\mu\nu}= \sum_{\substack{\eta\in\mathcal{P}\\ \eta_1\leq n-1\\|\eta|\equiv|\mu|\pmod n}} \sum_{\sigma\in S_n}\sum_{\lambda\in nQ+\sigma(\rho)-\rho} \epsilon(\sigma)q^{\frac{1}{2n}(\lambda|\lambda+2\rho)} \frac{1}{(q)_{m(\eta)}} \genfrac{[}{]}{0pt}{}{m(\eta)}{\frac{|\eta|-|\nu|}{n}(1^n)+\nu} \genfrac{[}{]}{0pt}{}{m(\eta)}{\frac{|\eta|-|\mu|}{n}(1^n)+\mu-\lambda}.$$

This is presented as a conjectural explicit expression for Kostka polynomials in terms of root-lattice sums and Gaussian multinomials; its resolution is not specified in the supplied text.