Periodic triangle sequences and algebraic powers

Let $n$ be a positive integer and let $\bigtriangleup$ denote the region of ordered $n$-tuples used by the triangle algorithm. Let $0 \leq \alpha_n \leq \cdots \leq \alpha_1 < 1$ be an $n$-tuple whose triangle sequence is $(k,k,k,\ldots)$. Periodic triangle-sequence conjecture. Then

$$\alpha_j=\alpha_1^j$$

for $1\leq j\leq n$, and $\alpha_1$ is a root of

$$x^{n+1}+kx^n+x^{n-1}+\cdots+x-1=0.$$

Conversely, if $\alpha$ is the real root of this equation lying between zero and one, then $(\alpha,\alpha^2,\ldots,\alpha^n)$ has purely periodic simplex sequence $(k,k,k,\ldots)$. This gives an explicit algebraic description of the tuple associated with a constant periodic triangle sequence; the source also indicates that analogous results should hold for purely periodic sequences of the form $(ij,ij,ij,\ldots)$.