Now what if you were allowing conjectures about the rule and someone said “An N*N koan with 2N + 2 black stones will always be green”? 
(I guess in the “arbitrary conjectures” variants we really need to allow a “not sure” response, since it could get tricky even with more natural-sounding conjectures 
)
Edit: @shinuito quickly made this comment outdated by providing a counterexample to the conjecture:
I’m fine with playing by this rule as well.
My idea of allowing only one guess, was that people would discuss their thoughts among each other and try to come up with ways to test hypotheses. In other words, I was expecting more discussions like “I think X may be a rule” and someone might respond with “Actually, X-a and X-b would both fit, how can we test which one it is?”
Also, I was interested in seeing a practical example of the induction problem play out.
I also very much like the idea of your rule, so I think we should finish at least this ongoing game using that rule.
@ArsenLapin1 already did a very good job of trying and confirming some edge cases with the last sets of koans. Perhaps we can do just a couple more tests before comitting to a guess?
Well, I’m convinced!
(I wouldn’t make a very good scientist.)
Well, let’s put it to a vote:
Feel free to submit more koans in the meantime. If no objections are made before 2023-07-02T01:00:00Z, I will reveal whether the guess is correct.
I voted that I wish to guess the rule, not out of any particular confidence in it, but just because I don’t feel any motivation to keep playing until we know whether this guess is correct or not.
Here are White counterparts of koan 363:
I thought I had an alternative, “A string can be extended to touch all four sides without capturing.”
But then I noticed that 362, which is missing from the big list, invalidates it. I’m still a little worried that we haven’t tried a lot of more full boards but I don’t have any alternative ideas at this point.
EDIT: actually 362 is consistent with that alternative. But it looks like 357 is not, so I still don’t have anything.
It is interesting to consider how diagonal connectivity is somehow a complement to normal connectivity, in the sense that connecting opposite sides one way prevents connecting the other pair of sides the other way.
How about this one, just in case?
Heheh
Changed my vote back to NO! 
I feel I can close the poll 
So to spell it out more explicitly:
These two types of connections are mutually exclusive:
If one type of connection exists, the other one cannot be made.
So I think the proposed rule was equivalent to this one:
Green koans are those where at least one of Black or White can add stones to connect all four sides into a single chain according to the rules of Go, but ignoring captures.
For example, this is how Black can go-connect 357:
It’s nice that that way of stating it refers to the rules of Go instead of this diagonal connectivity, but it seems unnatural to ignore captures. On the other hand, if you allow captures, just about anything would be green since Black can just capture all the White stones and keep placing stones to make a connection.
I proposed 374 above to suggest various kinds of captures, and indeed it fails.
So I’m thinking there’s something we need to figure out about this game that relates it better to actual Go rules.
Since this is not a valid Go position, I will not colour it.
Right, I intentionally crossed out the number to try to make that clear 
So, what’s next?
How about these small changes to 362?
To probe something about captures or corners.
I curious about these two
Does removing these Black stones turn the position green?
EDIT:
