Negative-index root-amplitude bound for $k$-generalized Fibonacci polynomials

For $k\ge3$ and $n<0$, let $\zeta_{n,k}$ be the maximum value of $|x_{\rm root}|^k$ among the nonzero roots of $\mathcal{F}_{n,k}(x)$. Let $r_{n,k}$ denote the remainder associated with $n$ modulo $k$ as in the source.

Negative-index amplitude-bound conjecture. Numerical evidence indicates that

$$\zeta_{n,k}\le\left\lfloor\frac{|n|}{k}\right\rfloor.$$

The bound is attained when $r_{n,k}=1$, equivalently when $n=-sk$ with $s\ge1$. The slopes of the branches decrease monotonically as $r_{n,k}$ increases from $1$ through $k-1$; the lowest branch has $r_{n,k}=k-1$, while the branch with $r_{n,k}=0$ does not follow a simple pattern.

Thus the conjecture gives a tight uniform bound on the negative-index root amplitudes, together with a proposed ordering of the nonzero-remainder branches. The $r_{n,k}=0$ behavior remains unresolved by the stated pattern.