  We prove a relative of the Optimal (Type B)} Colored Tverberg theorem of
\v{Z}ivaljevi\'{c} and Vre\'{c}ica which modifies this results in two different
ways. (1) Our result is valid if the number of rainbow faces is $q= p^n-1$,
where $p$ is a prime. (2) The size of rainbow simplices satisfies the condition
$\vert C_i\vert \in \{2q-2, 2q+1\}$ while in the original theorem $\vert
C_i\vert = 2q-1$ for all $i$.
