Non-$\mathrm{\ell }$-distance-balanced generalized Petersen graphs $GP(n,3)$ and $GP(n,4)$

A connected graph $G$ of diameter $\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(G)\ge \mathrm{\ell }$ is $\mathrm{\ell }$-distance-balanced if $\mathrm{\mid }{W}_{xy}\mathrm{\mid }=\mathrm{\mid }{W}_{yx}\mathrm{\mid }$ for every $x,y\in V(G)$ with $d_G(x,y)=\mathrm{\ell }$, where $W_{xy}$ is the set of vertices of $G$ that are closer to $x$ than to $y$. We prove that the generalized Petersen graph $GP(n,3)$ where $n>16$ is not $\mathrm{\ell }$-distance-balanced for any $1\le \mathrm{\ell }<\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(GP(n,3))$, and $GP(n,4)$ where $n>24$ is not $\mathrm{\ell }$-distance-balanced for any $1\le \mathrm{\ell }<\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(GP(n,4))$. This partially solves a conjecture posed by Š. Miklavič and P. Šparl (Discrete Appl. Math. 244:143-154, 2018).