Asymptotic growth conjecture for homogeneous Golombic and Levine sequences

Let $w=(b_1^{m_1}\dots b_k^{m_k})$ be a word in reduced form. Call $w$ homogeneous if either $\{b_1,\dots,b_k\}$ or $\{-b_1,\dots,-b_k\}$ is a subset of $\mathbb N_0$, and either $\{m_1,\dots,m_k\}$ or $\{-m_1,\dots,-m_k\}$ is a subset of $\mathbb N$. Let $\gamma w$ and $\ell w$ denote the golombic and Levine sequences based at $w$, respectively, and let $\varphi=\frac12(1+\sqrt5)$ and $\kappa$ be Mallows's constant.

Asymptotic growth conjecture. The golombic sequence $\gamma w$ is either bounded or

$$\lim\limits_{n\to\infty}\frac{\gamma_{n+1}w}{\gamma_nw\,\cdot\,\gamma_{n-1}w}=\varphi-1,$$

and the Levine sequence $\ell w$ is either bounded or

$$\lim\limits_{n\to\infty}\frac{\ell_{n+1}w}{\ell_nw\,\cdot\,\ell_{n-1}w}=\kappa.$$

The conjecture describes the asymptotic ratios of unbounded Golombic and Levine sequences based at homogeneous words. The Levine assertion generalises a hypothesis expressed by Mallows; the bounded cases remain included as an alternative.