Random unipotent Sylow subgroups of groups of Lie type of bounded rank

In 2001 Liebeck and Pyber showed that a finite simple group of Lie type is a product of $25$ carefully chosen unipotent Sylow subgroups. Later, in a series of works it was shown that $4$ unipotent Sylow subgroups suffice. We prove that if the rank of a finite simple group of Lie type $G$ is bounded, then $G$ is a product of $11$ random unipotent Sylow subgroups with probability tending to $1$ as $\mathrm{\mid }G\mathrm{\mid }$ tends to infinity. An application of the result to finite linear groups is given. The proofs do not depend on the classification of finite simple groups.