I'm a beginner in Bruhat-Tits theory, and the following phenomenon makes me puzzled.

So let $$F$$ be a $$p$$-adic field, with $$\mathfrak{o}\supset \mathfrak{p}$$ its ring of integers and maximal ideal, and $$q$$ its residue field order. Let $$G$$ be a connected reductive group over $$F$$. We assume $$G$$ to be quasi-split, and fix a Borel $$B=TU$$, $$U$$ the unipotent radical of $$B$$ and $$T$$ a maximal torus, all over $$F$$. Then we let $$\Sigma=\Sigma(G,T_0)\subset X^*(T_0)_F$$ be the root system ($$T_0$$ the split component of $$T$$).

Let $$P$$ be a standard $$F$$-parabolic subgroup (i.e. $$P\supset B$$), with its Levi decomposition $$P=MN$$ ($$M$$ a Levi and $$N$$ the unipotent radical), and $$\bar{P}$$ its opposite parabolic. Then we write $$\Sigma(\bar{P})$$ (resp. $$\Sigma_{\mathrm{red}}(\bar{P})$$) for the set of roots (resp. reduced roots) appearing in the Lie algebra $$\bar{\mathfrak{n}}:=\mathrm{Lie}(\bar{N})$$.

Let $$I_G$$ be a "standard" Iwahori subgroup (I'm sorry that I have not yet understood the precise definition here, but roughly, $$I_G$$ should "match" $$B$$) with pro-unipotent radical $$I_G^+$$ (a notion which so far I didn't fully understand either; nor did I find a precise definition in the literature, for example, in Kaletha-Prasad's famous new book), and $$K_G$$ a special maximal compact subgroup of $$G(F)$$ containing $$I_G$$. The simplest example is: $$G=\mathrm{GL}(n)$$ and $$B$$ the standard upper triangular Borel. Then $$K_G=\mathrm{GL}(n,\mathfrak{o})$$, $$I_G$$ is $$\begin{bmatrix} \mathfrak{o} & \dots & \mathfrak{o}\\ & \ddots & \\ \mathfrak{p} & & \mathfrak{o} \end{bmatrix}$$ and $$I_G^+$$ is $$\begin{bmatrix} 1+\mathfrak{p} & \dots & \mathfrak{o}\\ & \ddots & \\ \mathfrak{p} & & 1+\mathfrak{p} \end{bmatrix}.$$ In this example it's very clear, and reasonable, that $$(*) \prod_{\alpha\in \Sigma_{\mathrm{red}}(\bar{P})} [U_\alpha\cap K_G:U_\alpha\cap I_G^+]^{-1} = q^{-\dim \bar{\mathfrak{n}}}$$ for any standard parabolic $$P$$, where $$U_{\alpha}$$ is the root subgroup associated to $$\alpha$$. After all, the pro-unipotent radical $$I_G^+$$ should "contain some $$\mathfrak{p}$$ at negative root subgroups", and $$K_G$$ should "contain some $$\mathfrak{o}$$".

However, some "strange" phenomenon appears when the group is not split. There are examples of non-split, quasi-split $$G$$ with $$P=MN$$ such that $$\quad \prod_{\alpha\in \Sigma_{\mathrm{red}}(\bar{P})} [U_\alpha\cap K_G:U_\alpha\cap I_G^+]^{-1} \neq q^{-\dim \bar{\mathfrak{n}}}.$$ This looks somewhat weird. Why would such phenomena exist? This seems to mean that the pro-unipotent radical in BT-theory is very different from the "usual" unipotent radical. How do experts think about this?