Lev–Freiman conjecture for restricted sumsets

Let $A$ be a set of $k>7$ integers such that $A\subseteq [0,l]$, $0,l\in A$, and $\gcd(A)=1$. The restricted sumset is

$$2^{\wedge}A=\\{a+a':a,a'\in A,\\ a\ne a'\\}.$$

Lev–Freiman conjecture. One has

$$|2^{\wedge}A|\geqslant \left\{\begin{array}{ll} l+k-2 &\text{ if } l\leqslant 2k-5, \\\\ 3k-7 &\text{ if } l\geqslant 2k-4. \end{array}\right.$$

This conjecture was remarked by V. F. Lev in personal communication with G. A. Freiman as the restricted-sumset analogue of Freiman's $2A$ theorem. Its resolution status is not specified in the supplied source context.