We present a numerical contribution to a newly proposed fluid--structure interaction (FSI) benchmark involving the motion and rebound of an elastic solid immersed in a viscous incompressible fluid. The benchmark is conceived as a simplified and computationally accessible variant of contactless rebound configurations, while retaining the key physical mechanisms associated with strong fluid--structure coupling and near-contact dynamics. Our contribution is based on a monolithic formulation within an updated full Arbitrary Lagrangian--Eulerian (ALE) framework. The method combines implicit time integration with adaptive geometric re-meshing to maintain mesh quality as the solid approaches the wall and thin lubrication layers develop. Re-meshing is performed repeatedly during the simulation, preserving the fluid--solid interface geometry and enabling robust computations without introducing artificial contact, penalty terms, or prescribed rebound laws. A key feature of the presented results is that the rebound of the solid arises naturally from the coupled fluid--structure system as a consequence of fluid incompressibility and no-slip boundary conditions, without any additional ad hoc modeling. This behavior, which is well documented in the literature, contrasts with alternative numerical approaches that rely on explicit rebound or impact laws. The numerical approach builds on geometric re-meshing strategies previously developed for simulations of contactless rebounds of elastic solids in fluids [1], but is here applied to a modified benchmark configuration. This design choice facilitates systematic comparison of numerical methods while remaining challenging from the perspective of stability, accuracy, and energy consistency. We report benchmark quantities of interest including rebound height, characteristic times related to contact and separation, forces acting on the solid, pressure response at the wall, and energy exchange between the fluid and the structure. The presented results provide reference data for the benchmark and demonstrate that ALE-based methods combined with adaptive re-meshing offer a stable and accurate tool for this class of FSI problems. [1] J. Fara, S. Schwarzacher, and K. Tůma, Geometric re-meshing strategies to simulate contactless rebounds of elastic solids in fluids, Computer Methods in Applied Mechanics and Engineering, 422 (2024), 116824.