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Robin Ryder pointed out to me that 3 is indeed the absolute minimum one could observe because of the block constraint (bon sang, mais c’est bien sûr !). The distribution of the series of 3 digits being independent over blocks, the theoretical distribution under uniformity can easily be simulated:

#uniform distribution on the block diagonal
sheik=rep(0,9)
for (t in 1:10^6){
group=length(unique(c(sample(1:9,3),sample(1:9,3),sample(1:9,3))))
sheik[group]=sheik[group]+1
}

and it produces a result that is close enough to the one observed with the random sudoku generator. Actually, the exact distribution is available as (corrected on May 19!)