The polynomial-form conjecture for lucky spots in parking functions

Let $n$ be a positive integer and let $j$ be a positive integer. A lucky spot is a parking spot occupied by a car whose preferred spot is that spot. Let $f_j(n)$ denote the polynomial appearing in the claimed formula, and let $r_j$ be a rational number. Lucky-spot enumeration conjecture. The number of parking functions where the $j$-th spot is lucky has the form

$$\tfrac{j+1}{2j}(n+1)^{n-1}-f_j(n)(n-j+1)^{n-j+1},$$

where $f_j(n)$ is a polynomial of degree $j-2$ with rational coefficients. In particular, the asymptotic probability that the $j$-th spot is lucky is

$$\frac{j+1}{2j}-r_je^{-j}.$$

The formulas are suggested by the explicitly computed cases $j=1,2,3,4,5$; the conjecture proposes a uniform polynomial form and corresponding asymptotic expression for every $j$.