One-corner conjecture for rearrangements of 1-Naples parking functions

Let $n$ be a positive integer, and let a parking preference be represented by its associated lattice-path diagram, with corners defined as turns of the path. Suppose that there is only one corner above the line $y=n-x$. One-corner conjecture. That parking preference and all of its rearrangements are elements of $PF_{n,1}$. This conjecture concerns the characterization and enumeration of rearrangements of decreasing $k$-Naples parking functions that remain $k$-Naples parking functions. The paper presents it as an open conjecture for the case $k=1$.