The stronger parking-function bijection conjecture

Let $\mathcal{A}_c$ denote the set of parking functions with composition $c$, let $\operatorname{dinv}(PF)$ and $\operatorname{ides}(PF)$ be the diagonal inversion number and inverse descent set, and let $\operatorname{diagword}(PF)$ be the diagonal word. The stronger parking-function bijection conjecture. For integers $b\leq a-1$, there exists a bijection

$$f:\mathcal{A}_{[b,a]}\cup\mathcal{A}_{[a-1,b+1]}\longleftrightarrow\mathcal{A}_{[a,b]}\cup\mathcal{A}_{[b+1,a-1]}$$

that increases $\operatorname{dinv}$ by exactly one, preserves $\operatorname{ides}$ and the diagonal word, and satisfies

$$f(\mathcal{A}_{[a-1,b+1]})\subset\mathcal{A}_{[a,b]},\qquad f^{-1}(\mathcal{A}_{[b+1,a-1]})\subset\mathcal{A}_{[b,a]}.$$

This strengthens the preceding conjecture by requiring preservation of the diagonal word and specifying the images of the two summands. It was motivated by computer data and theoretical considerations, and the source gives no resolution.