Følner-sequence density conjecture for sumsets in the integers

Let $A,B subseteq ewline \mathbb Z$ be nonempty, and let $\mathbf F=(F_n)_{n\in\mathbb N}$ be a Følner sequence of finite subsets of $\mathbb Z$, meaning that

$$\lim_{n\to\infty}\frac{|(F_n+t)\triangle F_n|}{|F_n|}=0$$

for every $t\in\mathbb Z$. For $C\subseteq\mathbb Z$, write

$$\underline{\mathrm{d}}_{\mathbf F}(C)=\liminf_{n\to\infty}\frac{|C\cap F_n|}{|F_n|},\qquad \mathrm{d}_{\mathbf F}(C)=\lim_{n\to\infty}\frac{|C\cap F_n|}{|F_n|}$$

when the latter limit exists. Følner-sequence density conjecture. If

$$\underline{\mathrm{d}}_{\mathbf F}(A+B)<\underline{\mathrm{d}}_{\mathbf F}(A)+\underline{\mathrm{d}}_{\mathbf F}(B),$$

then there is a $k\in\mathbb N$ such that

$$\mathrm{d}_{\mathbf F}(A+B)=\mathrm{d}_{\mathbf F}(A+B+k\mathbb Z).$$

The source explicitly says that this appealing generalization is false; the displayed example shows why the related cofinite conclusion cannot be retained, while the conjectured density conclusion is also not valid in the stated generality.