Positivity and numerator growth for tropical cluster recurrences

Let $(d_n)$ be a sequence satisfying

$$d_{n+N}+d_n=\max\left(\sum_{j=1}^{N-1}[b_{1,j+1}]_+d_{n+j},\sum_{j=1}^{N-1}[-b_{1,j+1}]_+d_{n+j}\right),$$

with initial conditions $d_1=-1$ and $d_2=\cdots=d_N=0$ up to shifting the index, and suppose that $(d_n)$ is not periodic. Let $\operatorname{N}_n$ denote the numerator in the associated Laurent representation. Tropical growth conjecture. Then (a) $d_n>0$ for all $n>N$, and (b) there is a constant $\widetilde{\mathrm{C}}>0$ such that

$$\deg\operatorname{N}_n\sim\widetilde{\mathrm{C}}d_n\qquad(n\to\infty).$$

These properties would relate denominator-vector growth to numerator degree growth and support the entropy analysis of the associated birational maps; the conjecture is not resolved in the supplied source.