The convergence of the factor $\gamma(P)$ in the Iwasawa decomposition

Let $$G$$ be a connected, reductive group over a $$p$$-adic field $$k$$, $$A_0$$ a maximal split torus of $$G$$, and $$P = MU$$ a parabolic subgroup with $$M$$ containing $$A_0$$. Let $$\overline{P} = M \overline{U}$$ be the opposite parabolic, and let $$K$$ be a maximal compact subgroup in good position relative to $$P$$ and $$A_0$$. Then we have $$G = PK$$, and for $$g \in G$$, we can write $$g = u_P(g)m_P(g)k_P(g)$$ with $$u_p(g) \in U, m_P(g) \in M$$, and $$k_P(g) \in K$$ (nonuniquely).

Let $$\delta_P = q^{\langle 2 \rho, H_M(-) \rangle}$$ be the modulus character for $$P$$. The map $$\overline{U} \rightarrow \mathbb C^{\ast}$$ given by $$\bar{u} \mapsto \delta_P(m_P(\bar{u}))$$ is well defined and continuous. In Waldspurger's writeup on Harish Chandra's unpublished notes on the Plancherel formula, he considers the integral

$$\gamma(P) = \int\limits_{\overline{U}}\delta_P(m_P(\bar{u})) \space d \bar{u}$$

Is it obvious that this integral converges? I considered the following example:

Example: $$G = \operatorname{SL}_2(\mathbb Q_p)$$, $$P = MU$$ the usual Borel, and $$\overline{U}$$ the usual opposite, and $$K = \operatorname{SL}_2(\mathbb Z_p)$$. From the Iwasawa decomposition, we can write (nonuniquely)

$$\bar{u} = \begin{pmatrix} 1 & \\ x & 1 \end{pmatrix} = \begin{pmatrix}1 & x^{-1} \\ & 1 \end{pmatrix} \begin{pmatrix} x^{-1} \\ & x \end{pmatrix} \begin{pmatrix} 0 & -1 \\ 1 & x^{-1} \end{pmatrix}$$

whenever $$x \in \mathbb Q_p - \mathbb Z_p$$. Thus $$\delta_P(m_P(\bar{u})) = |x|^{-2}$$ when $$x \not\in \mathbb Z_p$$, and $$1$$ for $$x \in \mathbb Z_p$$, giving us

$$\gamma(P) = \operatorname{meas}(\mathbb Z_p) + \sum\limits_{l=1}^{\infty} p^{-2l} \operatorname{meas}(p^{-l}\mathbb Z_p - p^{-l+1} \mathbb Z_p)$$

$$= 1 + \sum\limits_{l=1}^{\infty}p^{-2l}(p^l-p^{l-1}) < 1 + \sum\limits_{l=1}^{\infty} p^{-2l}(p^l) = 1 + \sum\limits_{l=1}^{\infty} p^{-l} < \infty$$

if we normalize $$\operatorname{meas}(\mathbb Z_p) = 1$$. Maybe there is some way to reduce to the case of $$\operatorname{Res} \operatorname{SL}_2$$ or $$\operatorname{SU}(3)$$ where calculations will go similarly?