Akiyama's finiteness-property conjecture for shift radix systems

Let $\mathcal{D}_d^{(0)}$ denote the set of parameters whose shift radix system has the finiteness property, and let $\mathcal{E}_d$ denote the corresponding contractivity region.

Akiyama's finiteness-property conjecture. For every $d\in\mathbb{N}$,

$$\mathcal{D}_d^{(0)}\subset\mathcal{E}_d.$$

The conjecture concerns the boundary of the contractivity region: a parameter with the finiteness property should be contractive. It was attributed in the survey to Akiyama et al.; no general proof is supplied.