In this question on math.stackexchange.com I have made two conjectures the first of which I have proved. The second has not been settled. I post it here to seek a proof.

Given a quadratic surd $\sqrt d$ where $d$ is a natural number and not a perfect square. $(c_i)_{i=1}^\infty$ is the sequence of convergents of the continued fraction of $x$. Let $r_i:=\frac{c_{i+1}-\sqrt d}{c_i-\sqrt d},\,\forall i\in\mathbf N$. Let $n$ be the period of the continued fraction. It has been shown $\exists \,l_r:=\lim_\limits{i\rightarrow\infty}r_{in+r},\, \forall r\in\{0,1,\cdots,n-1\}$. Is the following statement, suggested by a numerical experiment, true?

For $n\le 2$, $l_0=l_1$.

For $n\ge 3$, there exists at least two distinct $l_r$'s.