  The nature of the spin excitations in superconducting cuprates is a key
question toward a unified understanding of the cuprate physics from long-range
antiferromagnetism to superconductivity. The intense spin excitations up to the
over-doped regime revealed by resonant inelastic X-ray scattering bring new
insights as well as questions like how to understand their persistence or their
relation to the collective excitations in ordered magnets (magnons). Here, we
study the evolution of the spin excitations upon hole-doping the
superconducting cuprate Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ by disentangling the
spin from the charge excitations in the experimental cross section. We compare
our experimental results against density matrix renormalization group
calculations for a $t$-$J$-like model on a square lattice. Our results
unambiguously confirm the persistence of the spin excitations, which are
closely connected to the persistence of short-range magnetic correlations up to
high doping. This suggests that the spin excitations in hole-doped cuprates are
related to magnons -- albeit short-ranged.
