I had done some searching, but apparently not enough.
I suspect 32 is the smallest positive integer radius for which there is an integer quadrilateral counterexample (where the restrictions must be satisfied by all reorderings of the edges).
Some background ...
The problem arose as a question in my mind based on my resolution of a question by José Carlos Santos in the thread "Non-constructible numbers of degree 4".
In one of my solutions to the problem, I analyzed a cyclic decagon with two edges of length 2 and the remaining 8 edges of length 1. I was able to show that the radius of the circumscribed circle was a non-constructible (by straight edge and compass) number of algebraic degree 4 over the rationals.
I began to wonder if there were non-trivial examples of a cyclic polygon with integer sides for which the radius of the circumscribing circle was an integer.
I was able to fully resolve the question for n = 3.
For n = 4, I saw some trivial ways, and wondered if those were the only ways.
The conjectures I posed were intended to promote an exploration of those types of questions.
With your latest counterexample, all the conjectures previously stated in this thread are now dead.
I'm not sure if there is any sensible revision that maintains the spirit of those conjectures. I'll think about it.
In any case, thank for playing -- I hope it was fun.