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This paper presents a novel algorithm, based on model predictive control (MPC), for the optimal guidance of a launch vehicle upper stage. The proposed strategy not only maximizes the performance of the vehicle and its robustness to external disturbances, but also robustly enforces the splash-down constraint. Indeed, uncertainty on the engine performance, and in particular on the burn time, could lead to a large footprint of possible impact points, which may pose a concern if the reentry points are close to inhabited regions. Thus, the proposed guidance strategy incorporates a neutral axis maneuver (NAM) that minimizes the sensitivity of the impact point to uncertain engine performance. Unlike traditional methods to design a NAM, which are particularly burdensome and require long validation and verification tasks, the presented MPC algorithm autonomously determines the neutral axis direction by repeatedly solving an optimal control problem (OCP) with two return phases, a nominal and a perturbed one, constrained to the same splash-down point. The OCP is transcribed as a sequence of convex problems that quickly converges to the optimal solution, thus allowing for high MPC update frequencies. Numerical results assess the robustness and performance of the proposed algorithm via extensive Monte Carlo campaigns.