  The Planck or the quantum gravity scale, being $16$ orders of magnitude
greater than the electroweak scale, is often considered inaccessible by current
experimental techniques. However, it was shown recently by one of the current
authors that quantum gravity effects via the Generalized Uncertainty Principle
affects the time required for free wavepackets to double their size, and this
difference in time is at or near current experimental accuracies [1, 2]. In
this work, we make an important improvement over the earlier study, by taking
into account the leading order relativistic correction, which naturally appears
in the systems under consideration, due to the significant mean velocity of the
travelling wavepackets. Our analysis shows that although the relativistic
correction adds nontrivial modifications to the results of [1, 2], the earlier
claims remain intact and are in fact strengthened. We explore the potential for
these results being tested in the laboratory.
