Pisot eigenvalue conjecture for periods of the multidimensional slow continued fraction algorithm

Let $K$ be a real cubic field, and let $\alpha^{(0)},\alpha^{(1)},\alpha^{(2)}$ be its positive $\mathbb{Q}$-basis. Set

$$\alpha=\frac{\alpha^{(1)}}{\alpha^{(0)}+\alpha^{(1)}+\alpha^{(2)}},\qquad \beta=\frac{\alpha^{(2)}}{\alpha^{(0)}+\alpha^{(1)}+\alpha^{(2)}}.$$

Let $\{\varepsilon_n\}_{n=0}^{\infty}$ be the expansion of $(\alpha,\beta)$. If $\varepsilon_{k+1},\ldots,\varepsilon_{k+l}$ is the period of this expansion, then Pisot eigenvalue conjecture. The matrix product

$$M_{\varepsilon_{k+1}}\cdots M_{\varepsilon_{k+l}}$$

has a Pisot number as an eigenvalue.

The conjecture is one of two claims introduced as supported by numerical experiments. The supplied text gives no proof or resolution.