Miklavič–Šparl conjecture on non--distance-balanced generalized Petersen graphs

Let $k\ge 2$ be an integer, and define

$$n_k=\begin{cases} 11, & k=2,\\ (k+1)^2, & k\text{ is odd},\\ k(k+2), & k\ge 4\text{ is even}. \end{cases}$$

For a graph $G$, write $D$ for its diameter. Miklavič–Šparl conjecture. For every $n>n_k$, the generalized Petersen graph $GP(n,k)$ is not $\ell$-distance-balanced for any integer $\ell$ with $1\le \ell<D$. Moreover, $n_k$ is the smallest integer with this property. The conjecture concerns the intermediate distance levels of generalized Petersen graphs: although the cited work proved the assertion for $k=2$, the general case is presented here as a conjecture. The minimality assertion specifies the sharp threshold in $n$ for each value of $k$.