Foata transform conjecture for spots of unit interval parking functions

Let $\alpha$ be a unit interval parking function, let $F$ denote the Foata transform, and let $\operatorname{spot}_\alpha$ denote the outcome permutation of $\alpha$. Foata transform conjecture. For any unit interval parking function $\alpha$,

$$\operatorname{spot}_{F(\alpha)}=F(\operatorname{spot}_\alpha).$$

The conjecture concerns how the Foata transform acts on the outcome of a parking function. It has been verified in the source for all unit interval parking functions of length at most $8$, but remains open in general.