We present indirect constraints on the stellar initial-mass-function (IMF) in nine massive elliptical galaxies with σ ≈ 300 km/s, via a comparison of dynamical and stellar-population based stellar masses. We use adaptive-optics assisted, high resolution kinematical data from the SINFONI Search for Supermassive Black Holes that allow us to constrain the dynamical stellar mass-to-light ratio in the very centre of each galaxy. Hence we measure the IMF in a galaxy region where the stellar mass dominates over dark matter, minimising any potential degeneracy between the two mass components. In six of our galaxies – those which have depleted stellar cores – we find an IMF consistent with the one measured in the Milky-Way via direct star counts. The three remaining, power-law galaxies have instead stellar masses about a factor of two times larger than expected from a Milky-Way type IMF, indicating either a more bottom-heavy IMF (like, e.g., the Salpeter IMF) or a dark-matter distribution that is degenerate with the stellar mass down to the very centres of these galaxies. The bottom-light IMF in our core galaxies is surprising in view of previous studies that suggested a systematic IMF variation where early-type galaxies with σ ≈ 300 km/s have a Salpeter or even more dwarf-dominated IMF. Core galaxies are particularly important since their unique central orbital structure offers an independent crosscheck for the dynamical models. Our models with a bottom-light IMF are consistent with the distribution of orbits predicted by SMBH-binary core-formation models. This indicates that spatially well resolved central kinematical data are important for determining unbiased dynamical stellar mass-to-light ratios. Our results imply either that the IMF in massive galaxies varies over a wider range than previously anticipated, and is not the same in core and power-law ellipticals, or else that there are systematic variations in the distribution of dark matter among massive early-type galaxies.