If not a log-normal distribution, what distribution describes this data?

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I have some empirical data which appeared to be best fit to a log-normal, but upon making a histogram of log(x) with 50 bins I got this result which appears to have some skew:

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What is the proper distribution to use for data whose log looks like the plot above? Some sort of "log-normal with skew"? How would I fit the data in this case?

Note: I believe this question is a similar one, but with no answer yet.