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Joint routing-and-scheduling has been considered in wireless mesh networks for its significant performance improvement. While existing work assumes it, accurate traffic information is usually not available due to traffic dynamics, as well as inaccuracy and delay in its measurement and dissemination. In addition, the joint routing and scheduling usually requires a centralized controller to calculate the optimal routing and scheduling and distribute such policies to all the nodes. Thus, even if the accurate traffic information is always available, the central controller has to compute the routing and scheduling repeatedly because the traffic demands change continuously. This leads to prohibitive computation and distribution overhead. Therefore, in this paper, we propose a joint routing-scheduling scheme that achieves robust performance under traffic information uncertainty. In particular, it achieves worst-case optimal performance under a range of traffic conditions. This unique feature validates the use of centralized routing and scheduling in wireless mesh networks. As long as the traffic variation is within the estimation range, the routing and scheduling do not need to be recomputed and redistributed. Through extensive simulations, we show that our proposed scheme meets the objective (i.e., optimizes the worst-case performance). Moreover, although it only guarantees the worst-case performance in theory, its average performance is also good. For example, our proposed scheme can perform better than a fixed optimal routing and scheduling scheme in more than 80 percent of 500 random traffic instances. Our scheme provides insights on the desired properties of multipath routing, namely, spatial reuse and load balancing.