We determine the homotopy nilpotency of $p$-localized $SU(n)$ when $p$ is a quasi-regular prime in the sense of M. Mimura and H. Toda, "Cohomology operations and homotopy of compact Lie groups I," Topology 9 (1970), 317–336. As a consequence, we see that it is not a monotonic decreasing function in $p$.
"Homotopy nilpotency in localized $SU(n)$." Homology Homotopy Appl. 11 (1) 61 - 79, 2009.