The golden-ratio limit conjecture for generalized diagonal Wythoff Nim

Fix $p,q\bin \mathbb{N}$, and let $a=a(p,q)$ and $b=b(p,q)$ denote the associated sequences. A pair $(p,q)$ is a splitting pair if it is a Wythoff pair or a dual Wythoff pair. The golden-ratio limit conjecture. The limit

$$\lim_{n\in \mathbb{N}}\frac{b_n}{a_n}$$

exists and equals the golden ratio $\Phi$ if and only if $(p,q)$ is a non-splitting pair. This predicts that splitting pairs are exactly the exceptional parameters for the golden-ratio asymptotic ratio.