The Haglund–Morse–Zabrocki parking-function bijection conjecture

Let $PF$ be a parking function, let $\operatorname{area}(PF)$, $\operatorname{dinv}(PF)$, and $\operatorname{ides}(PF)$ denote its area, diagonal inversion number, and inverse descent set, and let $\mathcal{A}_c$ be the set of parking functions with composition $c$. Write

$$c=[c_1,\ldots,c_k]=[c',c_i,c_{i+1},c”],$$

where $c'$ and $c”$ are the parts before and after $c_i,c_{i+1}$. The Haglund–Morse–Zabrocki bijection conjecture. If $c_i\leq c_{i+1}-1$, there exists a bijection

$$f:\mathcal{A}_{[c',c_i,c_{i+1},c”]}\cup\mathcal{A}_{[c',c_{i+1}-1,c_i+1,c”]}\longleftrightarrow\mathcal{A}_{[c',c_{i+1},c_i,c”]}\cup\mathcal{A}_{[c',c_i+1,c_{i+1}-1,c”]}$$

that increases $\operatorname{dinv}$ by exactly one and preserves $\operatorname{ides}$ and $\operatorname{area}$. This bijection would realize the commutativity relations predicted by the Haglund–Morse–Zabrocki conjectures at the level of parking functions; the source gives no resolution, so the claim remains open.