Rule of mutual exclusion: is the converse true?

There is a "famous" rule in spectroscopy,1 that goes this way:



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> If a compound is centrosymmetric, then its normal vibrational modes cannot be simultaneously IR and Raman active.
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and this is simple enough to prove: for a vibrational mode to be IR active, it has to transform under the same irrep as (one of) the cartesian axes $x$, $y$ or $z$, which are all *ungerade*. For a vibrational mode to be Raman active, it has to transform under the same irrep as one of $(x^2, y^2, z^2, xy, yz, xz)$, and since $u \otimes u = g$, these irreps cannot be the same as the irreps of $(x,y,z)$. Hence, a vibrational mode, which transforms under a single irrep, cannot simultaneously satisfy both conditions at once.


However, I am curious about the converse statement. 



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> If a compound does not have any vibrational modes which are simultaneously IR and Raman active, is it *necessarily* centrosymmetric?
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If the answer is *yes*, I would like a proof; if it is *no*, a counterexample, please.


(Just in case anybody is confused, because I am: A counterexample would be a molecule *without* a centre of inversion, which has no vibrational mode which is simultaneously IR and Raman active.)




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1 Famous enough for me to know, at least.




Have a look at the point groups O (not $\ce{O\_h}$) I, $\ce{D\_{5h} and C\_{5h}}$. A pic of an O point group molecule is shown below. The $\ce{c\_2}$ axes are not shown. Taken from molecule-viewer.com web site. The molecule is 'vanadium hexaoxo phosphonato' or more properly $\ce{C8H24O30P8V^+Cl^-}$, octakis(μ 3-methylphosphonato)-hexaoxo-penta- vanadium(v)-vanadium(iv) chloride clathrate !!!


[![vanadium O point group](https://i.stack.imgur.com/hEmNem.png)](https://i.stack.imgur.com/hEmNem.png)


Edit: I checked a bit more and found ; $\ce{ S8 }$


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If there is a molecule of $O$ (as opposed to $O\_h$) or $I$ (as opposed to $I\_h$) symmetry, it would be such a counter example (no mode that is both IR and raman active, and not centrosymmetric). 


<http://symmetry.jacobs-university.de/cgi-bin/group.cgi?group=903&option=4>


<http://symmetry.jacobs-university.de/cgi-bin/group.cgi?group=905&option=4>


Apparently a [snub cube](https://en.wikipedia.org/wiki/Snub_cube) is an example of the $O$ point group and in [Chemistry for the 21st Century](https://books.google.com/books?id=GQpieBwhxbcC&pg=PA145&dq=conforms%20to%20a%20snub%20cube&hl=en&sa=X&ved=0ahUKEwiNpLiR57fPAhXDJCYKHZl0By8Q6AEIJTAB#v=onepage&q=conforms%20to%20a%20snub%20cube&f=false) it is stated:



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> 9.5.2.5 Snub Cube (6) We have recently demonstrated the ability of six resorcin[4]arenes and eight water molecules to assemble in apolar media to form a spherical molecular assembly which conforms to a snub cube (Fig. 9.3)
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