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Recently, multi-infeed line-commutated converter-based high-voltage direct currents (LCC-HVDCs) represent one of the most attractive ways to accommodate large-scale power transfer in main load centers. However, this approach deteriorates the dynamic var reserve (DVR) in the receiving-end power grid, especially in scenarios with heavy loads. If the system does not reserve sufficient dynamic var in advance, a severe voltage dip may occur and further lead to consecutive commutation failures, ultimately resulting in DC blocking and voltage collapse. To address the voltage/var control challenges, a robust DVR assessment method is proposed in this paper. This method is used to determine both the inductive and capacitive DVR and prevent cascading failures. First, continuous dynamic algebraic equations are formulated using a piecewise functional block approximation method for efficient application of second-order trajectory sensitivity analysis (SOTSA). Next, a robust DVR assessment model is formulated, and the corresponding computational strategy is introduced to effectively and accurately solve the large-scale problem. Finally, in numerical simulations, the computational efficiency and accuracy of the proposed method are verified using a simple system and operational data from a real system in the Eastern China Power Grid.