The Fine-number enumeration conjecture for valleyless-tieless parking functions

A valleyless-tieless parking function of length $n$ is a parking function $(a_1,\ldots,a_n)$ with no index $1\leq i\leq n-1$ such that $a_i=a_{i+1}$ and no index $2\leq i\leq n-1$ such that $a_{i-1}>a_i<a_{i+1}$. Let ${\mathrm{VTPF}}_n$ denote the set of valleyless-tieless parking functions of length $n$, and let $F_i$ be the $i$th Fine number. Fine-number enumeration conjecture. Valleyless-tieless parking functions of length $n$ are enumerated by

$$|{\mathrm{VTPF}}_n|=F_{n-3}.$$

This is presented as an open problem in the paper; the statement includes the indexing convention for the Fine numbers but gives no evidence of a resolution.