Conjecture on generalized Fibonacci growth rates below $2^n$

Let $M$ be an $\omega$-categorical structure with growth rate $f_M(n)$, and for $k\in\mathbb{N}$ let the generalized Fibonacci numbers be defined by

$$F_{n,k}=\sum_{i=1}^{k}F_{n-i,k},$$

with $F_{n,k}=0$ for $n<0$ and $F_{0,k}=1$.

Generalized Fibonacci growth conjecture. If

$$f_M(n)<\frac{2^n}{p(n)}$$

for every polynomial $p(n)$, then there is some $k\in\mathbb{N}$ such that

$$\lim_{n\to\infty}F_{n,k}^{1/n}=\lim_{n\to\infty}f_M(n)^{1/n}.$$

The claim aims to determine the possible exponential growth constants in the range just below $2^n$; the source describes this subcase as more approachable and records the assertion as still conjectural.