Spherical Iwasawa decomposition conjecture for the hyperbolic Kac--Moody group

Let $G$ be the hyperbolic Kac--Moody group, let $K$ be its relevant compact subgroup, let $T^{ad}_{\mathbb R}$ be the real adjoint torus, and define $U_{i,\infty}^\pm=U_W^\pm U_i$ for $1\leq i\leq 2$. Spherical Iwasawa conjecture. The group $G$ admits the decomposition

$$G=KT^{ad}_{\mathbb R}U_{i,\infty}^\pm.$$

This is the proposed spherical analogue of an Iwasawa decomposition for the hyperbolic Kac--Moody group. The supplied text does not provide a proof or resolution, so the conjecture is left open.