The Rickard complex of a braid with strands colored by positive integers is a chain complex of singular Soergel bimodules. The complex determines the colored triply-graded homology and colored sl(N) homology of the braid closure, when closure is color-compatible. For each braid on two strands with any colors, we construct a minimal complex that is homotopy equivalent to its Rickard complex. It is not obtained by laborious simplification; instead, it is defined directly by explicit formulas obtained by educated guesswork and reverse engineering a conjectural connection between colored sl(N) homology and SU(N) representations of the knot group.