Maximum vertex-periodicity conjecture for fractal graphs

Let $K$) be one of the fractal graphs $SG$, $HG$, $PG$, $MG$, or $SGC$, and let

$$m=\min\{i:v\in K_i\}.$$

For a vertex $v$, its maximum periodicity is respectively $4(3^m)$ for $SG$, $2(3^m)$ for $HG$, $6(5^m)$ for $PG$, and $6(7^m)$ for both $MG$ and $SGC$.

Maximum periodicity conjecture. The maximum periodicity of any vertex $v$ in each listed fractal graph is given by the corresponding value above.

The paper presents these periodicities as observations for the listed fractal graphs; no proof or resolution is supplied in the source.