Convolution estimates and the number of disjoint partitions

Let $X$ be a finite collection of sets. We count the number of ways a disjoint union of $n-1$ subsets in $X$ is a set in $X$, and estimate this number from above by $\mathrm{\mid }X{\mathrm{\mid }}^{c(n)}$ where $$c(n)={(1-\frac{(n-1)\ln (n-1)}{n\ln n})}^{-1}\mathrm{.}$$ This extends the recent result of Kane-Tao, corresponding to the case $n=3$ where $c(3)\approx 1.725$, to an arbitrary finite number of disjoint $n-1$ partitions.