Conjecture on bounded generation of noncocompact lattices in SL(3)

Let $\Gamma$ be a lattice in either $\operatorname{SL}(3,\mathbb{R})$ or $\operatorname{SL}(3,\mathbb{C})$. A subgroup $U$ of $\operatorname{SL}(\ell,\mathbb{C})$ is unipotent if it is conjugate to a subgroup of the upper triangular matrices with all diagonal entries equal to $1$. A matrix group is virtually boundedly generated by unipotents if there are unipotent subgroups $U_1,\ldots,U_r$ such that $U_1U_2\cdots U_r$ is a finite-index subgroup of the matrix group.

Bounded-generation conjecture. If $\Gamma$ is any noncocompact lattice in either

$$\operatorname{SL}(3,\mathbb{R})\quad\text{or}\quad\operatorname{SL}(3,\mathbb{C}),$$

then $\Gamma$ is virtually boundedly generated by unipotents.

The conjecture is presented as a potential extension of the Carter–Keller–Paige bounded-generation theorem for suitable lattices in $\operatorname{SL}(2)$; the source gives no resolution.