Looking for a minimal triple

I am looking for a minimal triple for a particular set of phonemes. By minimal triple, I mean three actual English words that differ in one and only one phoneme between them. Examples therefore include:

• phony, tony, pony
• swim, swam, swum
• bad, bat, bath

The three phonemes that must appear one apiece in each of the three words are: /ɑ, ɒ, ɔ/. I can get pairs, but no triples, and I wonder why.

Length doesn’t count, so for example /aː/ counts as /a/ and /ɔː/ counts as /ɔ/.

If such a triple exists, what is it?

If no such triple exists, could it? Or do the various mergers preclude a threefold phonemic distinction here?

These have all to be in the same accent, of course. And preferably a rhotic one, too, because I’m trying to figure out something in particular about phonemic perception in rhotic speakers of North America, and non-rhotic accents will not further that goal.

As a last resort, if the only such triple you can find is a non-rhotic one, then go ahead and offer that one, perhaps along with some suggested explanation why there “can’t” be an equivalent rhotic triple.

For RP, doesn't khan, con, corn work?

If you allow widely-known foreign foods, how about pawed, pod, pad, where pad is as in pad thai (and that one even works for rhotic accents).

I'm betting you can find this in a Scottish or Irish English accent. I'm no expert, but I would suggest looking at cat/caught/cot as a possible triple of those vowels. You'd have to find one that escaped the caught/cot merger, and that has a rather back variant of the BATH vowel. To wit:

cat /kɑt/
caught /kɒt/
cot /kɔt/

In addition, most Scottish and Irish English dialects are rhotic.

Do these work?

Sawed, sod and Sade. (Sade being the last name of Marquis de Sade.)

Bought, bot and Baht. (Baht being the currency of Thailand.)