No, most fundamentally because “the same path” is a meaningless statement. All movement is relative so there is really no way to say an empty patch of space is the same patch or not. Independent of reference to other objects we can’t define something’s movement against some kind of underlying reference frame of space itself. That simply doesn’t exist!
So the entire universe could be stationary on average or moving at any speed under that of light, and all frames of reference in between are equally valid. That said our Sun is orbiting the center of our galaxy and relative to that reference point and the universe surrounding us Earth is charting a path it won’t repeat.
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