  The last three decades have witnessed the surge of quantum gravity
phenomenology in the ultraviolet regime as exemplified by the Planck-scale
accuracy of time-delay measurements from highly energetic astrophysical events.
Yet, recent advances in precision measurements and control over quantum
phenomena may usher in a new era of low-energy quantum gravity phenomenology.
In this study, we investigate relativistic modified dispersion relations (MDRs)
in curved spacetime and derive the corresponding nonrelativistic Schr\"odinger
equation using two complementary approaches. First, we take the nonrelativistic
limit, and canonically quantise the result. Second, we apply a WKB-like
expansion to an MDR-inspired deformed relativistic wave equation. Both
approaches imply equivalent results for single-particle quantum mechanics.
Based on a map between our approach and the generalized uncertainty principle
(GUP), we recognise in the latter the MDR which is least amenable to low-energy
experiments. Consequently, importing data from time-delay measurements, we
constrain the linear GUP up to the Planck scale and improve on current bounds
to the quadratic one by 17 orders of magnitude. MDRs with larger implications
in the infrared, however, can be tightly constrained in the nonrelativistic
regime, from which we use the ensuing deviation from the equivalence principle
to bound some MDRs to up to one order of magnitude below the Planck scale,
while constraining those customarily associated with the bicrossproduct basis
of the $\kappa$-Poincar\'e algebra to energy scales beyond $10^{15}$GeV.
