Let $G$ be the subgroup of integer matrices in $\mathrm{SO}(3,2)$.

(The invertible linear maps from a $5$ dimensional real vector space to itself which leave invariant a nondegenerate symmetric billinear form of signature $(3,2)$, and have determinant $1$.)

$G$ is a group of real rank $2$, so conjecturally, it should be boundedly generated.

(There exists an $m \in \mathbb{N}$, and cyclic subgroups $C_1, \dots, C_m \leq G$ such that $G = C_1 \cdots C_m$.)

I would like to know whether the following much weaker statement holds:

There exists a positive integer $m$, and finitely generated subgroups $H_1, \dots, H_m \leq G$ of infinite index, such that $G = H_1 H_2 \cdots H_m$.