Experimental and numerical studies of spatiotemporal femtosecond soliton propagation over up to 1 km spans of parabolic graded-index fibers reveal that initial multimode soliton pulses naturally and irreversibly evolve into a single-mode soliton. This is carried by the fundamental mode of the fiber, which acts as a dynamical attractor of the multimode system for up to the record value (for multimode fibers) of 5600 chromatic dispersion distances. This experimental evidence invalidates the use of variational approaches, which intrinsically require that the initial multimode propagation of a self-imaging soliton is indefinitely maintained.