Weight-preserving bijection between queue inversion and HHL statistics

Let $\lambda$ be a partition and $n$ a positive integer, and let $\operatorname{Tab}(\lambda,n)$ denote the set of tableaux of shape $\lambda$ with entries bounded by $n$. For fillings $\sigma$ and $\tau$, write $\sigma\sim\tau$ when they are row-equivalent, meaning that each corresponding pair of rows contains the same entries with the same multiplicities. Let $\operatorname{maj}$, $\operatorname{inv}$, and $\operatorname{quinv}$ denote the major index, HHL inversion, and queue inversion statistics.

Bijection conjecture. Given $\lambda$ and $n$, there exists a bijection

$$\delta:\operatorname{Tab}(\lambda,n)\longrightarrow\operatorname{Tab}(\lambda,n)$$

such that, for every $\sigma$, $\delta(\sigma)\sim\sigma$, $\operatorname{maj}(\delta(\sigma))=\operatorname{maj}(\sigma)$, and

$$\operatorname{inv}(\delta(\sigma))=\operatorname{quinv}(\sigma).$$

Such a bijection would be sufficient to prove the tableau formula for the modified Macdonald polynomial from the HHL formula, because it would preserve row content and major index while exchanging queue inversion and HHL inversion weights. The supplied text gives no evidence that this conjecture has been resolved.