Event selection is central to high-intensity particle physics experiments to allow DAQ systems to manage the corresponding data rates. This is true for the MUonE experiment, which will be composed of 40 tracking stations and targets to reconstruct elastic scatters of muons on atomic electrons. Since the signal process and some backgrounds can be distinguished using track information alone, online track reconstruction can result in lower data rates with higher purity. Online track-fitting and vertexing for event selection will be implemented directly on FPGAs, using High-Level Synthesis to convert C++ code into an HDL description. This contribution will focus on results from a test beam at the M2 beamline at CERN in 2023, subsequent improvements to the algorithm, and plans for a 2025 test beam in the same location. In 2023, the standalone reconstruction of tracks from high-rate muons was performed, and the algorithm showed good agreement with offline reconstruction. Later developments yielded significant timing and resource use improvements, which allow for single track fit results to be calculated above 50 MHz. This was achieved through pipelining the algorithm and implementing custom linear algebra functions which include basic assumptions on the fits. These results demonstrate the feasibility of FPGA-based online reconstruction for event selection for the MUonE experiment. In the 2025 test beam, a more general reconstruction algorithm will be demonstrated. This includes reconstructing events with higher occupancy, and implementations of candidate track selection and vertexing. Expected milestones and planned improvements to the more general algorithm will also be described. Finally, it will be shown that this algorithm and the assumptions made to improve performance can be generalized to applications beyond MUonE.