Hmm. The article is technically correct but irrelevant. The case where necessity fails relies on three conditions: (1) the number of voters is even (2) the number of voters is small (3) at least one voter has their optimal preferences exactly identical to the proposed equilibrium; not merely 'very close' but exactly. All three (plus some additional, complicated conditions) must hold for Plott's conditions to be sufficient but not necessary.

(2) is obviously not a concern here, for nation-state electorates. (3) is implausible: just introduce a suitably fine-grained continuum of possible policies. If you still have an ideal voter at the equilibrium, it's not fine-grained enough.

On (3), in particular: in general, mainstream economics ignores degenerate cases in utilitarian analysis. That's why the additional conditions are not mentioned: it requires that of a (finite) number of voter ideal points, at least one of them must fall on the equilibrium. But in a multidimensional phase space, the set of equilibrium points is a set of measure zero! Why would you care about that case?