  In this article, we show that if $KG$ is Lie nilpotent group algebra of a
group $G$ over a field $K$ of characteristic $p>0$, then $t_{L}(KG)=k$ if and
only if $t^{L}(KG)=k$, for $k\in\{5p-3, 6p-4\}$, where $t_{L}(KG)$ and
$t^{L}(KG)$ are the lower and the upper Lie nilpotency indices of $KG$,
respectively.
