The concentration conjecture for coefficient-lattice minima of four-element sets

Fix an integer $h\ge3$ and a real number $\epsilon>0$. For a four-element subset $A$ of $[n]=\{1,2,\dots,n\}$, let $h_1$ and $h_2$ denote the first and second successive $L^1$-minima of its coefficient lattice.

Concentration conjecture. There is an integer $n_0$ such that, for every $n\ge n_0$, at least $(1-\epsilon)\binom{n}{4}$ of the four-element subsets of $[n]$ have $h_1>h$; among those with $h_1\le h$, the proportion having $h_2\le h$ is at least $1-\epsilon$.

The conjecture is intended to explain the concentration of four-element sumset sizes around values governed by the first two coefficient-lattice minima. The paper describes the relevant proportions as not yet proved, so the asymptotic assertion remains open.