  The ground state of half-doped manganites involves intricate spin, charge and
orbital orderings, which are difficult to discern experimentally. In this work,
we resort to the theoretical analysis of the spin fluctuation spectrum of the
half-doped bilayer Pr(Ca$_{0.9}$Sr$_{0.1}$)$_2$ Mn$_2$O$_7$ in order to get an
insight of its electronic ground state. By means of the linear spin wave
approximation, we compute the magnon dispersion for a family of localized spin
models, which can describe several phases proposed for the ground state of
half-doped manganites, like the intermediate one proposed by Efremov et al.
[Nat.Mats. 3, 853, (2004)] along with its particular cases corresponding to
Goodenough's CE phase [Phys. Rev. 100, 564, (1955)] and Zener polaron or dimer
phases. We obtain an excellent agreement between theory and experiment when the
ground state is assumed to be a generalized Goodenough's CE phase, with a
Mn-charge disproportionation inside the experimentally expected range. As
essential ingredients for our improved fit of the upper and lower magnon
branches measured around the gap, we identified two next-nearest neighbour
exchange interactions between the planar Mn zig-zag chains, one for each type
of Mn ion present. In connection with this finding, we revisited the magnetic
excitations of the laminar related compounds, focusing on the upper magnon
branches. Here we prove that their measurement would provide the key to
identify unambiguously the ground state present in the layered half-doped
manganites.
