The splitting conjecture for generalized diagonal Wythoff Nim

Fix $p,q\in\mathbb{N}$, let $\pi=\pi_{p,q}$, and let $a=a(p,q)$ and $b=b(p,q)$ be the associated sequences. Let $U$ and $\mathbb{N}_0$ have the meanings fixed earlier in the paper, and let splitting and splitting precisely once be as defined for sequences of integer pairs. The splitting conjecture.

1. The sequence

$$((n,\pi(n)))_{n\in U}=((a_n,b_n))_{n\in\mathbb{N}_0}$$

splits if and only if $(p,q)$ is a splitting pair.

2. If $(p,q)$ is a splitting pair, then $((n,\pi(n)))_{n\in U}$ splits precisely once.

The conjecture characterizes splitting through the Wythoff and dual Wythoff pairs and predicts that every splitting case has exactly one split.