  Majorana bound states (MBSs) located at the ends of a hybrid
superconductor-semiconductor nanowire are only true zero modes if their
characteristic localization length is much smaller than the nanowire length,
$\xi_M\ll L$. Otherwise, their wave function overlap gives rise to a finite
energy splitting that shows a characteristic oscillatory pattern $\sim
e^{-2L/\xi_M}\cos(k_F L)$ versus external parameters that modify the Fermi
momentum $k_F$. Detecting such "Majorana oscillations", measurable through
low-bias conductance, has been proposed as a strategy for Majorana detection in
pristine nanowires. Here we discuss how this detection scheme does not work in
full-shell hybrid nanowires, an alternative design to partial-shell nanowires
in which a superconductor shell fully wraps the semiconductor core. Using
microscopic models, we provide both numerical simulations for Al/InAs hybrids
as well as analytical approximations in terms of general nanowire parameters.
We find that Majorana oscillations with flux in full-shell nanowires are absent
in a wide portion of parameter space. This absence is not a signature of
non-overlapping left- and right-end MBSs, but a consequence of the Majorana
oscillation period being systematically larger than the flux window of odd
Little-Parks lobes where Majorana zero-energy peaks are predicted to appear.
Our results demonstrate that split near-zero modes or individual zero-energy
crossings should not be dismissed as trivial even if they are found not to
oscillate with flux.
