When I studied the conjecture 1, I found a generalization as follows:
Conjecture: Let $A_iB_iC_iD_i$ for $i=1,2,...,n$ be $n$ squares in a plane, with $A_i \rightarrow B_i \rightarrow C_i \rightarrow D_i \rightarrow A_i$ all counter clockwise, (or all clockwise) for $i=1,2,...,n$. Then show that: $$Area(A_1A_2...A_n)+Area(C_1C_2...C_n)=Area(B_1B_2...B_n)+Area(D_1D_2...D_n)$$
See the conjecture in GeoGebra
