Eureka! It fits a Pearson type: 3 distribution
N. C. Matalas, James R. Wallis · Water Resources Research · 1973
Under the assumption that a random variable is distributed as Pearson type 3, a comparison was made between moment and maximum likelihood estimates of the parameter values of the distribution and the variate values at specified probability levels. For the region where maximum likelihood solutions may be obtained, maximum likelihood estimates yield solutions that are less biased and less variable than the comparable moment estimates. When these results are extended to quite small samples, say, N≈25, they become quite pronounced as the probability p becomes greater than N/(N+1).