  Numerous accidental near-degeneracies exist between the $2\nu_2$ and $\nu_4$
rotation-vibration energy levels of ammonia. Transitions between these two
states possess significantly enhanced sensitivity to a possible variation of
the proton-to-electron mass ratio $\mu$. Using a robust variational approach to
determine the mass sensitivity of the energy levels along with accurate
experimental values for the energies, sensitivity coefficients have been
calculated for over 350 microwave, submillimetre and far infrared transitions
up to $J\!=\!15$ for $^{14}$NH$_3$. The sensitivities are the largest found in
ammonia to date. One particular transition, although extremely weak, has a
sensitivity of $T=-16,738$ and illustrates the huge enhancement that can occur
between close-lying energy levels. More promising however are a set of
previously measured transitions with $T=-32$ to $28$. Given the astrophysical
importance of ammonia, the sensitivities presented here confirm that
$^{14}$NH$_3$ can be used exclusively to constrain a spatial or temporal
variation of $\mu$. Thus, certain systematic errors which affect the ammonia
method can be eliminated. For all transitions analyzed we provide frequency
data and Einstein A coefficients to guide future laboratory and astronomical
observations.
