Conjecture on diagonal positions in Wythoff star

Let $S_1,S_2,S_3,S_4\subseteq\mathbb{N}$ be the sets defined by

$$S_1=\{3,8,11,21,32\},\qquad S_2=\{129,362\},$$ $$S_3=\{x\in\mathbb{N}\setminus\{19\}\mid\text{the Zeckendorf coding of }x\text{ ends in }101001\},$$ $$S_4=\{x\in\mathbb{N}\mid\text{the Zeckendorf coding of }x\text{ ends in }1\}.$$

Diagonal Wythoff-star conjecture. The position $(i,i)$ belongs to $\mathcal{P}(\mathrm{W}^\star)$ if $i\in(S_1\cup S_4)\setminus(S_2\cup S_3)$, and belongs to $\mathcal{N}(\mathrm{W}^\star)$ if $i\in\mathbb{N}\setminus(S_1\cup S_2\cup S_3)$. Numerical data from computer simulations motivate this classification; the source gives no proof or resolution.