Conjecture on the asymptotic injectivity of the difference-multiset map

Let $A \subseteq \{0,1,2,\dots,n\}$ with $0 \in A$. Define $\phi$ on all such sets by sending $A$ to the multiset with $|A|^{2}$ elements given by

$$A-A=\{ x-y:x,y\in A \}.$$

Difference-multiset injectivity conjecture. As $n \to \infty$, the image of $\phi$ contains $2^{n-1}+o(2^{n})$ distinct elements. For a set $A$ with maximum element $a'$, the reflected set $A'=\{a'-a:a\in A\}$ has the same difference multiset, so the conjecture asserts that, asymptotically, a difference multiset almost always determines the pair $A$ and $A'$. Establishing this conjecture would imply that almost all relevant 0,1-polynomials have an irreducible non-reciprocal part. The paper presents it as an open problem associated with a lemma, and does not provide a resolution.