Multi-EGS-groups: exponent of congruence quotients

Given a multi-EGS-group $K$ acting on the $p$-adic rooted tree, where $p$ is any prime number, we compute the exponent of the congruence quotient $K_n=K\mathrm{/}{\text{St}}_K(n)$ for all $n\ge 1$. The formula that we obtain for $\exp (K_n)$ only depends on $p$, $n$ and the periodicity of $K$.