Distribution of the sum of reciprocal parts for distinct parts partitions

Given an integer partition of $n$ into distinct parts, the sum of the reciprocal parts is an example of an egyptian fraction. We study this statistic under the uniform measure on distinct parts partitions of $n$ and prove that, as $n\to \mathrm{\infty }$, the sum of reciprocal parts is distributed away from its mean like a random harmonic sum.