Eventual monotonicity of recursively defined sequence growth

Let $f\in\mathbb Q(d_1,d_2)$ be a rational function defined on $\mathbb Z^+\times\mathbb Z^+$ and suppose

$$f(d_1,d_2)>0\quad\forall\,d_1,d_2\in\mathbb Z^+,$$

and

$$\lim_{d_1,d_2\longrightarrow\infty}\left(f(d_1,d_2)\bigg/\bigg(\frac{d_1d_2}{d}\bigg)^k\right)\in\mathbb R^+$$

for some $k\in\mathbb R^+$. If $n_d$ is a sequence recursively defined by the source's recurrence and $n_1>0$, then the sequence $\sqrt[d]{n_d}$ is eventually increasing: there exists $d^*\in\mathbb Z^+$ such that

$$\sqrt[d]{n_d}\le\sqrt[d+1]{n_{d+1}}\qquad\forall\,d\ge d^*.$$

This conjecture concerns eventual log-convex-type behavior of recursively generated sequences and is motivated by asymptotic patterns in Gromov–Witten invariants. Its precise scope depends on the recurrence referenced by the source, which is not included in the supplied context.