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 $\left|G\right|$ 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.