  The study of existence of a universal $C^*$-completion of the $^*$-algebra
canonically associated to a Hecke pair was initiated by Hall, who proved that
the Hecke algebra associated to $(\operatorname{SL}_2(\Qp),
\operatorname{SL}_2(\Zp))$ does not admit a universal $C^*$-completion.
Kaliszewski, Landstad and Quigg studied the problem by placing it in the
framework of Fell-Rieffel equivalence, and highlighted the role of other
$C^*$-completions. In the case of the pair $(\operatorname{SL}_n(\Qp),
\operatorname{SL}_n(\Zp))$ for $n\geq 3$ we show, invoking property (T) of
$\operatorname{SL}_n(\Qp)$, that the $C^*$-completion of the $L^1$-Banach
algebra and the corner of $C^*(\operatorname{SL}_n(\Qp))$ determined by the
subgroup are distinct. In fact, we prove a more general result valid for a
simple algebraic group of rank at least $2$ over a $\mathfrak{p}$-adic field
with a good choice of a maximal compact open subgroup.
