Bosma–Dekking–Steiner metallic-mean iteration conjecture

For an integer $j\geq 1$, let $\alpha_j$ be the $j$-th metallic mean, so that $\alpha_j^2=j\alpha_j+1$. Let $r_1$ be the unique positive real number such that all terms defined by

$$r_{n+1}=\frac{n^2}{r_n-jn+1}$$

are positive. Bosma, Dekking, and Steiner's conjecture. The iterative sequence $\lfloor r_n\rfloor$ equals the Beatty sequence

$$\left\lfloor \alpha_j n-\frac{1+\alpha_j}{2\alpha_j-j}\right\rfloor.$$

This conjecture concerns extending the known iterated-function description from the displayed example to all metallic means; its resolution is not given in the supplied text.