Conjecture on Fibonacci-numeration P-positions of Wythoff star

Let $F_n$ denote the Fibonacci numbers, and let $(A_i,B_i)$ be the pairs occurring in Theorem 2. For all $ne 3$ and every $i$ such that the stated inequalities hold, the following positions are $P$-positions:

$$A_i+B_i\le F_{2n-4}$$

implies

$$(F_{2n-1},F_{2n}-1-A_i-B_i),$$

and

$$A_i\le F_{2n-4}$$

implies

$$(F_{2n-1}+A_i,F_{2n}-1).$$

Conjecture on Fibonacci-numeration P-positions. For all $n\ge 3$ and all $i$ satisfying these conditions, both displayed positions are $P$-positions of the game under consideration. This would extend the preceding characterization of Wythoff-star positions by similar methods; the source provides no resolution.