Generic horofunction boundary conjecture for infinite Sierpinski polygon graphs

Let $\Gamma_\xi$ be an infinite Sierpinski polygon graph that grows away from the vertex $j$. Define

$$S:=\{s\in V_r\mid s\neq j\text{ and }sj\text{ occurs infinitely many times in }\xi\}.$$

Generic horofunction boundary conjecture. If $S$ is empty, the horofunction boundary is as in the stated theorem: it has exactly two Busemann points and countably many non-Busemann points. If $s\in S$, then there are two Busemann points; moreover, in each of the three cases

$$j+\frac{r-f(r)}{2}\leq s\leq j+\frac{r+f(r)}{2},\qquad s<j+\frac{r-f(r)}{2},\qquad s>j+\frac{r+f(r)}{2},$$

there exist countably many non-Busemann points, and the three cases produce different non-Busemann points.

This is proposed as a generalization of the preceding theorem from the special sequence $\xi=wj^\infty$ to generic sequences defining graphs that grow away from $j$. The parser supplies no evidence that the proposed generalization has been proved or disproved.