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I think it's very misleading, the values in max_stat$stats are not the distribution of the entire original dataset, it is probably just the subset of the values tested, there is perhaps some logic in maxstat::maxstat.test which specifies which values to test, it probably does not test every possible cut point, it might only test up to the lowest such that there are at least n in each group.
This is a very easy fix.
maxstat::maxstat.test$stats <- data[,var_i] # this will copy all rows of var_i into stats then the distribution and plots will align more with the original data.