Can we observe the shapes of MOs? Condensed question formulation:

Is there an experimental method to directly visualise the 3D form of a MO wave function, or at least the electron density associated with it?

Full statement:
Molecular orbital theory is a crowning achievement of modern Chemistry: it is by far the most accurate description of chemical bonding we possess and accounts for the vast majority of related phenomena.
An array of evidence exists to confirm predictions from MO theory. Ab initio computational methods such as Hartree-Fock are able to produce a very accurate description of orbital coefficients and energies, which match experimental data for the energies (via photoelectron spectroscopy etc.) or predicted reaction mechanisms/FMO interactions (in kinetic runs).
But do we have any experimental evidence for the shape of MOs?
I'd presume the most feasible way of observing some measure related to the MO wave function $\Psi$ would be to follow a probabilistic approach and try imaging the electron density assumed to be given by $|\Psi|^2$.
Of course, X-ray diffraction first springs to mind; the notorious issue with that being the phase problem. In my superficial understanding: due to the loss of phase information, in going from the XRD pattern to the electron density isosurface (via ab initio methods), essentially we model some initial guess phase and try to fit it to the intensity data.
Since usually the location of nuclei is of far greater practical interest, the formalisms developed to model the phases (e.g. Hansen-Coppens) tend to focus on charge densities localised on each atom (though they are not necessarily of spherical symmetry). This is useful in actually getting the form of the molecule, but produces a picture different to the expected electronic occupation of delocalised MOs.
Is it possible to instead produce the delocalised picture if our ab initio phase guess is based on a RHF calculation? But even if possible, does this have any more legitimacy than the localised picture? I.e. it seems, as far as XRD is concerned, these are equally possible interpretations of the same intensity data where we just forced some phase information as convenient.
Sorry for the long description, this is an in depth presentation of my thoughts on the issue thus far. Any experimental technique/fundamental aspect of MO behaviour I'm missing would be appreciated. But the bottom line is, as stated in the beginning, can we experimentally visualise the shape of MOs or at least their modulus squared?
The very question you pose is addressed thoroughly in this open access work:
https://www.nature.com/articles/ncomms9287 . 
The short answer to your question is that the electron density can be mapped using a technique akin to diffraction as described in their work.  You mentioned a distinction between the electron density and the orbitals, and the orbital is not itself observable; but as you suggested, the electron density can be mapped.