  Let $\mathbf{G}$ be a connected reductive algebraic group over a $p$-adic
local field $F$. In this paper we study the asymptotic behaviour of the trace
characters $\theta _{\pi}$ evaluated at a regular element $\gamma $ of
$\mathbf{G}(F)$ as $\pi$ varies among supercuspidal representations of
$\mathbf{G}(F)$. Kim, Shin and Templier conjectured that
$\frac{\theta_{\pi}(\gamma)}{{\rm deg}(\pi)}$ tends to $0$ when $\pi$ runs over
irreducible supercuspidal representations of $\textbf{G}(F)$ with unitary
central character and the formal degree of $\pi$ tends to infinity. For
$\textbf{G}$ semisimple we prove that the trace character is uniformly bounded
on $\gamma$ under the assumption, which is expected to hold true for every
$\textbf{G} (F)$, that all irreducible supercuspidal representations of
$\textbf{G}(F)$ are compactly induced from an open compact modulo center
subgroup. Moreover, we give an explicit upper bound in the case of $\gamma $
ellitpic.
