If the answer is "it depends", what does it depend on? Does convergence depend on the ratio of predictor variables to sample size, or the size of $R^2$, or something else?
I am mainly interested in CIs on Unadjusted $R^2$, but would also be interested to hear the answer to this question in relation to CIs on Adjusted $R^2$.
Primarily I'm interested in finding out the conditions under which convergence happens or doesn't happen, but I would also appreciated an explanation of why.
EDIT: I am at present using the calculator made here: http://www.danielsoper.com/statcalc3/calc.aspx?id=28
I'm interested to know whether this formula gets close to nominal coverage with larger sample size (or stays the same, or gets further away from nominal coverage). I'm also interested to know if this holds true for all $R^2$ CIs or just ones calculated using this formula.