Weak Weyl's law for congruence subgroups

Let $G$ be a connected and simply connected semisimple algebraic group over $\mathbb{Q}$ and let $\mathrm{\Gamma }\subset G(\mathbb{Q})$ be an arithmetic subgroup. Let $K_{\mathrm{\infty }}\subset G(\mathbb{R})$ be a maximal compact subgroup and let $d$ be the dimension of the symmetric space $G(\mathbb{R})\mathrm{/}{K}_{\mathrm{\infty }}$. Let $\sigma$ be an irreducible unitary representation of $K_{\mathrm{\infty }}$. We prove that for every $\mathrm{\Gamma }$ there exists a normal subgroup ${\mathrm{\Gamma }}_1\subset \mathrm{\Gamma }$ of finite index such that the quotient of the counting function of the ${\mathrm{\Gamma }}_1$-cuspidal spectrum of weight $\sigma$ and $T^{d\mathrm{/}2}$ has a positive lower bound as $T\to \mathrm{\infty }$.