  We discuss the reconstruction of generic 3+1-dimensional space-time
geometries from covariant quantum spaces as backgrounds in the IKKT matrix
model. An explicit recipe to realize generic classical geometries is provided.
Even though this typically entails some higher-spin contributions, these do not
significantly modify the physical content of the model in the weak gravity
regime. This justifies the framework for emergent gravity given by the
semi-classical matrix model, supplemented by an induced Einstein-Hilbert action
which arises in the presence of fuzzy extra dimensions.
