TL;DR: Nobody's sure, and none of the proposals are universally accepted.
It's pretty clear that maneō came from the PIE root *m-n-, which has plenty of cognates across the IE world. But if I understand right, that's not what you're asking about.
The consonants come straight from PIE, and the -e- of the second conjugation is well-understood. The real mystery is the -a- in the middle. Most often, -a- comes from the laryngeal *h₂ (or *h₁ or *h₃ in Italic), but there's no sign of a laryngeal in this root.
The paper you've linked gives a number of different ways this might have developed, from big names like Sihler, De Vaan, and others. You've already read what they have to say on the matter; I'll add my own slightly more far-fetched theory.
Latin man- may perhaps have come from the zero-grade stem, *mṇ-. In Latin, this would have regularly developed into *men- or *min-, via (according to Meiser, in Klein's Handbook) some sort of intermediate *mẽ-.
But in the closely-related Sabellic languages, Meiser's *ẽ instead becomes an in stressed (i.e. initial) syllables: compare Oscan fanguam "tongue (acc)" against Latin linguam (<*dẽgwām), or Oscan anter "between" against Latin inter (<*ẽter).
So it's entirely possible that the Latin *men- was affected, or possibly even replaced, by the Sabellic *man-. This is attested with other words, such as lupus "wolf" (instead of the expected *lucus), which shows the Sabellic shift of *kʷ to p instead of the Latin *kʷ → k / u_.
Unlikely? Perhaps—I can't find any actual attestation of a Sabellic verb *man-, though the sound changes involved would be regular. But I've never seen a more regular explanation than this: they all require some sort of irregularity (in this case, the Sabellic influences). So I have to agree with Leppänen: "the current scholarship is not conclusive enough to show which ablaut grade Lat[in] manē- actually continues, and which changes it has undergone."