Unstable homotopy theory 3 — Suppose X is a simply connected finite complex.

Suppose X is a simply connected finite complex. Do the Steenrod reduced powers P^t act trivially on the mod p cohomology of the loop space of X when p is sufficiently large? This question is apparently due to Wilkerson, and is taken from Chuck McGibbon's problem list on phantom maps, in Stable and unstable homotopy, Fields Institute Communications 19 (a book published by the AMS), where there are other questions as well.