Great. Now, show us the evidence that this experiment has ever occurred. Or, are we still talking about a made-up hypothetical thought experiment? If that is the case, then I can just make up any outcome I want as well.
No getting through to you then...
Look, the thing is that you can ALWAYS look south from ANY location in the southern hemisphere at ANY time stars are visible in that location and see the southern pole star, also referred to as Sigma Octantis.
This is FACT.
You going on and on about all 3 locations needing to see the stars at the same time and wanting perfect experiments is just your excuse for not wanting to believe it. Actually I think you already know this to be a fact and you are simply trolling with bigger and bigger demands.
It is not an experiment, it is an observation anyone can make and has made throughout history.
You complained and I gave you 3 far away locations where they can all be in astronomical twilight and see stars at the same time.
You complained again and I gave you 3 locations that were all in nighttime at the same time as you requested. Salvador, Cape Town and Perth.
Now again you complain and it feels like you wont stop complaining until I give up because you are requesting something unreasonable.
You know, instead of making an unreasonable request, with a small timespan, requiring people in 3 locations where all locations need great weather to view the stars, the request nearly impossible. Try thinking a bit. You might realize that it is way WAY easier to disprove this observation, by simply showing any night at any time of night in the southern hemisphere where you can't see Sigma Octantis. Or show that Sigma Octantis would move during the night.
This however is impossible for you, since it is fact that Sigma Octantis is so close to the south pole that you might as well call it stationary from our spinning earth.
TLDR; Lets put it another way. If in fact we could do this experiment and show Sigma Octantis being visible from all locations at the same time all visible due south... What would you demand next? Or would you admit that we can all see the same stars in the southern hemisphere?