The eventual non-$\ell$-distance-balance conjecture for generalized Petersen graphs

Let $GP(n,k)$ be the generalized Petersen graph, let $d$ denote graph distance, and let $D$ be its diameter. For an integer $k\geq 2$, define

$$n_k=\begin{cases}11,&k=2,\\(k+1)^2,&k\text{ odd},\k(k+2),&k\geq4\text{ even}. \end{cases}$$

The eventual non-$\ell$-distance-balance conjecture. For any $n>n_k$, the graph $GP(n,k)$ is not $\ell$-distance-balanced for any integer $\ell$ with $1\leq\ell<D$. Moreover, $n_k$ is the smallest integer with this property. The conjecture formalizes computational evidence that, for each fixed $k$, sufficiently large generalized Petersen graphs are diameter-distance-balanced but fail to be $\ell$-distance-balanced at every smaller distance.