Ellipse conjecture for the roots of the Fibonacci-level characteristic polynomial

Let $f_n(\lambda)$ be the polynomial appearing in the paper, let $F_{n+1}$ denote the $(n+1)$st Fibonacci number, and consider the roots of

$$f_n(\lambda)=F_{n+1}.$$

Ellipse conjecture. The roots of $f_n(\lambda)=F_{n+1}$ lie on an ellipse.

This conjecture is motivated by numerical plots of the roots, which appear to fit an ellipse. The supplied text gives no proof or resolution.