Sroysang's conjecture on the growth rate of Fibonacci functions with period k

Let $f$ be a real-valued Fibonacci function with period $k\in\mathbb{N}$, meaning that

$$f(x+2k)=f(x+k)+f(x)$$

for every $x\in\mathbb{R}$. Let $\phi$ denote the golden ratio. Sroysang's conjecture.

$$\lim_{x\rightarrow\infty}\left\{\frac{f(x+k)}{f(x)}\right\}=\phi.$$

This conjecture concerns the asymptotic exponential growth rate of Fibonacci functions with period $k$. The result in the paper proves a more general statement using the continued-fraction expansion of the root of a nonsquare integer.