Algebraicity from eventually periodic triangle sequences

Let $(\alpha_1,\ldots,\alpha_n)$ be an $n$-tuple of real numbers in $\bigtriangleup$, and suppose that its triangle sequence is eventually periodic. Algebraicity conjecture for periodic triangle sequences. Each $\alpha_j$ is algebraic of degree at most $n$, and

$$\dim_{\mathbf Q}\mathbf Q[\alpha_1,\ldots,\alpha_n]\leq n.$$

This is a multidimensional analogue of the principle that periodic continued-fraction data characterize algebraic numbers. The statement concerns the consequences of eventual periodicity for the individual degrees and for the joint field generated by the coordinates; no resolution is supplied in the source.