They're all three of them from the Latin suppono (supponeresupposuisuppositum) 'to place under; bury; substitute (falsely), counterfeit; add, annex; subject'.
Yep. Here is what the OED says of the etymology of suppository: "[ad. late L. suppositrium, neut. sing., used subst., of suppositrius placed underneath or up, f. supposit-, suppnre to SUPPONE. Cf. F. suppositoire.]"
On the outskirts of Buenos Aires, a city rife with monuments, when I lived there a long time ago, there was an enormous tank for storage of propae gas. It was shaped like a two-ended bullet and probably held 100,000 gallons. It was locally known as "El Monumento Al Supositorio."
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Ah 'spose (the correct pronunciation, of course) that if one uses a suppository long enough, one may operate under the supposition that emissions will take place. That would be a connection, rather more direct than etymological.
But it passes all knowledge whether wordnerds deal in such directness, doth it not?