Minimum number of clues to fully specify any sudoku?

We know from this paper that there does not exist a puzzle that can be solved starting with 16 or fewer clues, but it implies that there does exist a puzzle that can be solved from 17 clues. Can all valid sudoku puzzles be specified in 17 clues? If not, what is the minimum number of clues that can completely specify every valid puzzle? More formally, does there exist a valid sudoku puzzle (or, I guess it would be a set of puzzles) that cannot be uniquely solved from only 17 clues? If so, then what is the minimum number of clues, $C$, such that every valid sudoku puzzle can be uniquely specified in $C$ or fewer clues?

The fewest clues required for a proper Sudoku is 17, but not all completed grids can be reduced to a proper 17 clue Sudoku. About 49,000 unique (non-equivalent) Sudokus with 17 clues have been found. (A proper Sudoku has only one solution).

The most clues in a minimal Sudoku is believed to be 40 (two are known to exist), but it has not been proven if this is the maximum. (minimal means that if any clue is removed, the Sudoku would have more than one solution, and therefore not be a proper Sudoku)

(This information is from Wikipedia, of which these statements are well referenced).

StackExchange Computer Science Q#163, answer score: 5