When I studied the conjecture 2, I found a generalization:
Conjecture: Let $A_{i1}A_{i2}...A_{in}$ for $i=1,2,..,n$ be a regular n-gons, such that $A_{i1} \rightarrow A_{i2} \rightarrow ....\rightarrow A_{in} $ all counter clockwise, (or all clockwise). Let $G_j$ be the centroids of n-polygon $A_{1j}A_{2j}...A_{nj}$ for $j=1,2,...,n$. Then show that: $G_1G_2...G_n$ is a regular n-gon. In the figure as attachted n=6.
A generalization: Let $A_{i1}A_{i2}...A_{in}$ for $i=1,2,..,n$ be $n$ similar n-gons, such that $A_{i1} \rightarrow A_{i2} \rightarrow ....\rightarrow A_{in} $ all counter clockwise, (or all clockwise). Let $G_j$ be the centroids of n-polygon $A_{1j}A_{2j}...A_{nj}$ for $j=1,2,...,n$. Then show that: $G_1G_2...G_n$ form a $(n+1)$ th similar n-polygon.
Put $\omega:=e^{2\pi i/n}$. Then there are complex numbers $z_k$ and $c_k$ with $$A_{kj}=z_k+c_k\omega^j\quad(1\leq k\leq n, \ 1\leq j\leq n)\ ,$$ and your "averaged polygon" has vertices $$G_j:={1\over n}\sum_{k=1}^n A_{kj}={1\over n}\sum_{k=1}^n z_k+\left({1\over n}\sum_{k=1}^n c_k\right)\>\omega^j\ .$$ This shows that the "averaged polygon" is again regular.
The "generalization" can be handled similarly: Begin with $$A_{kj}=z_k+c_k w_j\quad(1\leq k\leq n, \ 1\leq j\leq n)\ .$$ Here $z_k$ is the centroid of the $k^{\rm th}$ polygon, the $c_k$ are complex stretching factors, and the $w_j$ with $\sum_{j=1}^n w_j=0$ determine the common shape of the polygons.