Kimberling’s golden-ratio array conjectures

Let $\varphi=(1+\sqrt{5})/2$ and, for each integer $n\ge 1$, define the row $R_n=\{\lfloor k\varphi^n\rfloor:k\ge 1\}$. Kimberling's conjectures are: (1) for every $n\ge 1$, $R_{2n-1}\cap R_{2n}=\varnothing$; (2) if the entries of $R_{2n-1}\cup R_{2n}$ are arranged in increasing order and each entry is replaced by its rank in this union, then the entries originating in $R_{2n-1}$ form the lower Wythoff sequence $\{\lfloor k\varphi\rfloor:k\ge 1\}$, while those originating in $R_{2n}$ form the upper Wythoff sequence $\{\lfloor k\varphi^2\rfloor:k\ge 1\}$; and (3), defining $a(N)=\#\{n\ge 1:N\in R_n\}$ for each positive integer $N$, every positive integer occurs infinitely often among the values of $a$, that is, $\forall v\ge 1\ \exists$ infinitely many $N\ge 1$ such that $a(N)=v$.