Freiman–Lev conjecture on restricted double sumsets

Let $A=\{a_0,a_1,\ldots,a_{k-1}\}$ be a set of $k>7$ integers such that

$$0=a_0<a_1<\cdots<a_{k-1}$$

and let $d(A)=1$ denote that the greatest common divisor of all differences between elements of $A$ is $1$. Write $2^{\wedge}A$ for the restricted twofold sumset, consisting of sums of two distinct elements of $A$.

Freiman–Lev conjecture. One has

$$\left|2^{\wedge}A\right|\geq \begin{cases} a_{k-1}+k-2, & a_{k-1}\leq 2k-5,\\ 3k-7, & a_{k-1}\geq 2k-4. \end{cases}$$

This conjecture concerns the minimum possible size of a restricted double sumset in terms of the largest element of a normalized integer set. The source attributes it to Freiman and Lev; no resolution is supplied here.