Number of sudoku puzzles vs valid chess positions

Question

The basic question is - which one is bigger? This possibly needs some clarification:
By a sudoku puzzle I mean a grid with some cells filled with numbers and others empty so that they can be filled to a complete, valid sudoku grid in a unique fashion. We consider two puzzles the same if they have the same solutions and blanks in the same places. The obvious upper bound is $10^{81}$ as there are $81$ cells which can be either empty or be the number from $1$ to $9$. This can be easily improved to $2^{81} \cdot 6,671,248,172,291,458,990,080 \approx 6 \cdot 10^{46}$ - the number of subsets of all valid sudoku grids. $2^{81} = 2417851639229258349412352$ can be replaced by: $${81 \choose 17} + {81 \choose 18} + \dots + {81 \choose 81} = 2417851595207450142980773$$ As we know that a sudoku must have at least $17$ clues to have a unique solution. As the difference is only minor the estimate doesn't change.
Number of chess positions seems to be more tricky. The upper bound seems to be something around $10^{52}$ (at least according to https://en.wikipedia.org/wiki/Shannon_number) so it might be possible that the number of chess positions is bigger. We will assume that the positions are the same if all their pieces are in the same places, the special move privileges are the same and there is the same player to move. We will ignore $50$ move rule counter and threefold repetitions as it would ramp up this number massively ($50$ move rule only by about $50$ but for the latter we should count for every position so far how many times it has occured).
Is there some proven lower bound that will show that the number of chess positions is bigger than $6 \cdot 10^{46}$?