Limit-point conjecture for the generalized Rogers-Ramanujan continued fraction

Let $a=\zeta_k$ and $x=\zeta_m$ be primitive roots of unity, with $m\geq3$. Limit-point conjecture. The continued fraction $K_a(x)$ converges exactly when

$$\sqrt{\frac14+a^m}\notin\mathbb Q(a,x)\quad\text{and}\quad \frac14+a^m\notin\mathbb R_{\leq0}$$

and hence exactly when $5\nmid m$ or $k\nmid m$, and $a^m\neq-1$. In the convergent case its limit is the one given by the paper's general limit formula. If $5\mid m$ and $k\mid m$, the two limit points are

$$\frac{P_{m-2}(a,x)}{\frac12\pm\sqrt{\frac14+a^m}-ax^{m-1}P_{m-3}(a,x)}.$$

For $3\mid m$ and $a^m=-1$, these limit points may also occur; otherwise, the remaining subsequences produce exactly the three additional limit points specified by the source. This conjectural picture organizes convergence, divergence, and all predicted limit points of the continued fraction at primitive roots of unity.