In 1949 Shannon proved, using a non-constructive counting argument, that some boolean functions have exponential circuit complexity, see  and many texts on computational complexity. This result has been strengthened later, showing that, in the limit, almost all boolean functions have exponential circuit complexity. It is often said that, even in 2022, we still cannot construct a single example of such a function. I'm interested in, and slightly confused about, the state-of-the-art in research on this question. In particular:
Wikipedia  states that "complexity theorists have only been able to prove superpolynomial circuit lower bounds on functions explicitly constructed for the purpose of being hard to calculate" which seems to contradict that no explict function is know.
Are there any candidate boolean functions (or family of functions) that are conjectured to have exponential circuit complexity, but where this has not been proven?
What is the state of exhaustive exploration of the circuit complexity of $n$-ary boolean functions? Presumably there is some small $n$ such that this complexity has been computed for all $i<n$?
- C. E. Shannon, The synthesis of two-terminal switching circuits https://ia802700.us.archive.org/6/items/bstj28-1-59/bstj28-1-59.pdf
- Wikipedia entry Circuit complexity https://en.wikipedia.org/wiki/Circuit_complexity#History