Two-line approximation conjecture for queens in exile

Let $S_c$ be the row containing the exiled queen in column $c$, and let $\phi=(1+\sqrt{5})/2$ be the golden ratio. Two-line approximation conjecture. There are constants $\epsilon_1$ and $\epsilon_2$ such that

$$|S_c-c\phi|<\epsilon_1\quad\text{if }S_c>c,$$

and

$$\left|S_c-\frac{c}{\phi}\right|<\epsilon_2\quad\text{if }S_c<c.$$

Numerical computations show that the queen positions remain within bounded distance of two lines of slopes $\phi$ and $1/\phi$, but the paper does not establish the existence of such uniform bounds.