Using a fully transparent micropillar array chip and an optical "injection" setup capable of writing pulsed and continuous patterns into the flow by uncaging a fluorescent dye, highly detailed measurements of the transversal dispersion process in two-dimensional (2D) chromatographic beds could be made. With the use of water-glycerol mobile phase spanning a wide range of viscosities, the obtained data cover a broad range of reduced velocities (0 < ν < 3000) and show a clear leveling-off of the transversal dispersion coefficient at large reduced velocities. With dependence on the packing density, this leveling-off occurs at a value of about Dtrans/Dmol = 10 (ε = 0.4), Dtrans/Dmol = 4 (ε = 0.6) and Dtrans/Dmol = 2.5 (ε = 0.8). Another interesting observation that could be made is that the effect of the bed order on the observed transversal dispersion process is relatively small. The observed leveling-off in the relation between the measured Dtrans values and the reduced liquid velocity furthermore clearly invalidates the classical Galton-board model, predicting a linear increase of Dtrans with the reduced velocity. On the other hand, it corroborates a recently proposed series-connection transport model for Dtrans in 2D porous media.