The relationship between many-body interactions and dimensionality is integral to numerous emergent quantum phenomena. A striking example is the Bose gas, which upon confinement to one dimension (1D) obeys an infinite set of conservation laws, prohibiting thermalization and constraining dynamics. In our experiment, we demonstrate that such 1D behavior can extend deep into the three-dimensional (3D) crossover. Starting from a weakly interacting Bose gas trapped in a highly elongated potential, we perform a quench to instigate dynamics of a single density mode. Employing the theory of Generalized Hydrodynamics, we identify the dominant relaxation mechanism as the 1D dephasing of the relevant collective excitations of the system, the rapidities. Surprisingly, the dephasing remains dominant even for temperatures far exceeding conventional limits of one-dimensionality where thermalization should occur. We attribute our observations to an emergent Pauli blocking of 3D excitations, caused by the rapidities assuming fermionic statistics, despite the gas being purely bosonic. Thus, our study suggests that 1D physics is less fragile than previously thought, as it can persist even in the presence of significant perturbations. More broadly, by employing the exact Bethe ansatz solutions of the many-body system, we facilitate an interpretation of how the emergent macroscopic behavior arises from the microscopic interactions.