The inv-maj bijection for fillings and conjugate partition shapes

Let $\mathcal{F}$ be the set of all fillings of Young diagrams with positive integers. For a filling $\sigma$, let $\operatorname{inv}(\sigma)$ and $\operatorname{maj}(\sigma)$ denote its inversion and major-index statistics, let $\mathcal{F}_\mu^\alpha$ be the fillings of shape $\mu$ and content $\alpha$, and write $\mathcal{F}_\mu^\alpha|_{\operatorname{inv}=a,\operatorname{maj}=b}$ for those with the indicated statistic values. The inv-maj bijection conjecture. There is a natural isomorphism of weighted sets

$$\varphi:(\mathcal{F};\operatorname{inv},\operatorname{maj})\to(\mathcal{F};\operatorname{maj},\operatorname{inv})$$

which interchanges $\operatorname{inv}$ and $\operatorname{maj}$ and sends a partition shape to its conjugate; equivalently, for every $a,b,\mu,\alpha$, it restricts to a bijection

$$\varphi:\mathcal{F}_\mu^\alpha|_{\operatorname{inv}=a,\operatorname{maj}=b}\to\mathcal{F}_{\mu^\ast}^\alpha|_{\operatorname{inv}=b,\operatorname{maj}=a}.$$

This would provide an elementary combinatorial proof of the $q,t$-symmetry relation for transformed Macdonald polynomials, extending the classical inv-maj equidistribution on permutations. The source does not state a resolution.