The switch map's dinv and fixed-point conjecture

Let $\mathcal{P}_{3,n}$ be the set of $(3,n)$-parking functions, let $\operatorname{pides}(\pi)=2^a1^b$, let $\Phi$ be the involution whose fixed points are the fixed parking functions, and let $\mathbb{S}$ be the switch map. The switch-map conjecture. For any $\pi\in\mathcal{P}_{3,n}$ with $\operatorname{pides}(\pi)=2^a1^b$, (a)

$$\operatorname{dinv}(\pi)=\operatorname{dinv}(\mathbb{S}(\pi)).$$

(b) When $n$ and $3$ are coprime, if $\pi$ is a fixed point of $\Phi$, then so is $\mathbb{S}(\pi)$, and

$$\operatorname{pides}(\mathbb{S}(\pi))=2^b1^a.$$

The conjecture was verified for parking functions with at most 10 rows, while the text notes that part (b) fails when $n$ is a multiple of $3$.