Units in residue classes

Question

Let $K$ be a CM-field of degree $2n$. (Quadratic extension of totally real number field)

Let $\mathcal{O}_K$ be the ring of integers in $K$, and $m\geq 1$ an integer. Let $U_K$ be the group of units in $\mathcal{O}_K$.

I am interested in finding a lower bound of $T(m)$ in the following:

$$\pi_m: U_K \longrightarrow (\mathcal{O}_K/m\mathcal{O}_K)^{\times} $$

$$T(m)= |\textrm{Im}(\pi_m)|$$

I am particularly interested in lower bound of $c m^{\alpha}$ for some positive constants $c$, $\alpha$ that depend only on $K$.

Is it possible to have such bound?

Thank you.