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It seems to be folklore that every 5-dimensional convex polytope has a 3-gonal or 4-gonal face of dimension two. I was not able to track down a source for that claim.

Alternatively, I would be interested in a nice and short proof of the statement.

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Not an answer, but a possible lead. I found in

Grünbaum, Branko. Convex Polytopes. Vol. 221. Springer Science & Business Media, 2013. (MSN)

the following passage:

          Image from Grünbaum - Convex polytopes, referencing a paper of Kalai
          Grünbaum, p.224b.
Unfortunately I cannot access the reference [f], which is On low-dimensional faces that high-dimensional polytopes must have, by Gil Kalai. But perhaps this will help: @GilKalai.

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    $\begingroup$ The doi of the cited article is 10.1007/BF02122781 The result is in Theorem 1. $\endgroup$
    – efs
    May 24, 2020 at 21:42
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    $\begingroup$ Thanks to @EFinat-S, the entertainingly titled Kalai - On low-dimensional faces that high-dimensional polytopes must have (MSN). Sadly for this situation, @-ing GilKalai won't work, since they haven't yet been involved in the conversation. $\endgroup$
    – LSpice
    May 24, 2020 at 21:43
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    $\begingroup$ @LSpice: Thanks re @-ing. I did not understand that. I guess to protect users from spamming. $\endgroup$
    – Joseph O'Rourke
    May 24, 2020 at 21:51
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    $\begingroup$ The general reason for why anything with SO software is the way it is is that it is intended to fit only the exact needs and preferences of its creators. (For example, you can't @ from questions at all, so it wouldn't work even if GilKalai were here; you can only @ one user per comment, and it mustn't be the author of the parent post; etc.) Lots of such quirks are documented at meta.stackexchange.com/questions/43019/… . MO has some special treatment grandfathered in, but, as far as I know, nothing affecting @-replies. $\endgroup$
    – LSpice
    May 24, 2020 at 22:28
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    $\begingroup$ @LSpice I changed the link to a doi link, rather than a Springer one, since they are prone to breaking. And I included author and title in the plain text of the answer, for better information content/reference. $\endgroup$
    – David Roberts ♦
    May 24, 2020 at 22:30

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