Gullerud–Johnson–Mbirika conjectures on roots of Fibonacci-shifted path polynomials

Let $P_n$ be the path graph on $n$ vertices, let

$$f_n(\lambda)=\sum_{k=0}^{\lfloor n/2\rfloor}(-1)^{n+k}\binom{n-k}{k}\lambda^{n-2k}$$

be its characteristic polynomial, and let $F_0=0$, $F_1=1$, with $F_n$ denoting the Fibonacci numbers. For each $n$, let $R_n$ be the set of roots of $f_n(\lambda)=F_{n+1}$. Gullerud–Johnson–Mbirika conjectures. The following assertions hold: (1) for each $n$, the points in $R_n$ lie on an ellipse; (2) the maximum real part of a point in $R_n$ is unbounded as $n$ increases; (3) for all $n$, the imaginary parts of points in $R_n$ lie within $[-1,1]$; (4) if $n$ is even, then $R_n$ contains exactly two distinct real roots; and (5) if $n$ is odd, then $R_n$ contains exactly one real root, which is negative. These conjectures concern the geometric distribution of roots of characteristic polynomials associated with path graphs. The source reports partial progress toward two conjectures, an infinite class of polynomials contradicting a third, and evidence against a fourth; the supplied text does not identify which numbered assertions have each status.