We deal with the star problem in trace monoids which means to decide whether the iteration of a recognizable trace language is recognizable. We consider trace monoids K _n={a ₁ b ₁}* × … ×{a _n b _n }*. Our main theorem asserts that the star problem is decidable in a trace monoid M iff it is decidable in the biggest K n submonoid in M. Thus, future research on the star problem can focus on the trace monoids K n. The recently shown decidability equivalence between the star problem and the finite power problem [14] plays a crucial role in the paper.