Jerebic's eventual non-distance-balancedness conjecture for generalized Petersen graphs

Let $n\ge 3$ and $1\le k<n/2$. The generalized Petersen graph $GP(n,k)$ has vertex set

$$V(GP(n,k))=\\{u_i:i\in\mathbb{Z}_n\\}\cup\\{v_i:i\in\mathbb{Z}_n\\}$$

and edge set

$$E(GP(n,k))=\\{u_iu_{i+1}:i\in\mathbb{Z}_n\\}\cup\\{v_iv_{i+k}:i\in\mathbb{Z}_n\\}\cup\\{u_iv_i:i\in\mathbb{Z}_n\\}.$$

Jerebic's conjecture. For any $k\ge 2$, there exists a positive integer $n_0$ such that $GP(n,k)$ is not distance-balanced for every $n\ge n_0$. The conjecture was positively resolved by Yang et al.; in fact, if $k\ge 2$ and $n>6k^2$, then $GP(n,k)$ is not distance-balanced.