  We investigate conformal dimension for the class of infinite hyperbolic
groups in the Gromov density model $\mathcal{G}^d_{m,l}$ of random groups with
$m \geq 2$ fixed generators, density $0 < d < 1/2$ and relator length $l \to
\infty$. Our main result is a lower bound linear in $l$ at all densities $0 < d
< 1/2$ achieved by building undistorted round trees coming directly from lower
density Gromov random groups.
