Fibonacci limit

The Fibonacci sequence can be defined with induction as $F_n=F_{n-1}+F_{n-2}$, where $F_1=0$ and $F_2=1$. It is also well-known that the limit of the ratio between two consecutive terms $\lim \frac{F_{n+1}}{F_n}$ is the golden ratio $\phi$.

In the generalized Fibonacci sequence below, the original Fibonacci sequence is $F_n^2.$ $F_n^m=\sum_{i=1}^{m}{F_{n-i}^m}\ \text{ with }\ F_n^m=0\ \text{ if } \ n<m\ \text{ and }\ F_m^m=1.$ Every sequence will also have a generalized golden ratio $\phi_m=\displaystyle \lim_{n \rightarrow \infty}{ \frac{F_{n+1}^m}{F_{n}^m}}.$

What is the limit of the generalized golden ratio below? $\lim_{m \rightarrow \infty}{\phi_m} = \lim_{m \rightarrow \infty}{\lim_{n \rightarrow \infty}{\frac{F_{n+1}^m}{F_{n}^m}}}$