As a woodworker, I can say it depends on type of wood plus direction of grain..... and plywood would be more resilient than many types of solid wood. In the circumstances, DtG's solution should be quite adequate.

In a 6 foot+ length, one would have the grain parallel to the floor, thus subject to bending. Simply adding a thin plywood edge to the original piece..(on both sides, preferably)... would likely have served also. I imagine that 2 1X2s were probably cheaper than a 2X2 to manufacture.