Uniqueness of the upper transition attractor for bounded affine families

Let $F_t$ be a bounded one-parameter affine family with attractors $A_t$, and let $t_0$ be the threshold at which attractors cease to exist. A compact set $A^*$ is an upper transition attractor if it is the limit of a sequence $A_{t_k}$ with $t_k\to t_0$. The family $F_t$ is uniform if the map $t\mapsto A_t$ is uniformly continuous on $(0,t_0)$. Upper transition attractor conjecture. For a bounded one-parameter family $F_t$, the following three equivalent statements hold: there is a unique upper transition attractor; $\lim_{t\to t_0}A_t$ exists; and $F_t$ is uniform. The question concerns whether attractors have a unique limiting behavior at the threshold $t_0$. It is verified in dimension one, while graphical evidence supports it in dimension two; the general case remains open.