Short answer is “No”.
According to General Relativity, gravitational effects and acceleration are equivalent. Gravity is basically a distortion of space-time. Acceleration propels one into the future (vs the frame of reference they started in). The faster than light speed discussed here would, in effect, be a distortion of space-time. Why wouldn’t there be a time travel aspect? Faster than light also means, according to theory, and if ever possible, going BACK in time.
And... time dilation *does* occur within gravitational fields, more or less depending on the amount of mass involved. The space-time manipulation discussed here seems to me to be not too different than a gravitational field.