GRD monotonicity for insertion depth and tree height

Let $T_n(f)$ be the preferential attachment tree on $[n]$ generated by the nondecreasing attachment function $f:{\mathbb{N}}_0\to\mathbb{R}_{>0}$. Write $D_n(f)$ for the graph distance from the root to the newest vertex $n$, and $H_n(f)$ for the maximum graph distance from the root to any vertex. For nondecreasing functions $f,g:{\mathbb{N}}_0\to\mathbb{R}_{>0}$, say that $f$ dominates $g$ in growth-ratio order when

$$\frac{f(k+1)}{f(k)}\geq\frac{g(k+1)}{g(k)}$$

for every $k\in\mathbb{N}_0$, equivalently when $f(k)/g(k)$ is nondecreasing. Set ${\mathbb{E}}_f[D_n]:={\mathbb{E}}[D_n(f)]$ and ${\mathbb{E}}_f[H_n]:={\mathbb{E}}[H_n(f)]$. GRD monotonicity. If $f\succeq_{\mathrm{GR}}g$, then for all $n\geq1$,

$${\mathbb{E}}_f[D_n]\leq{\mathbb{E}}_g[D_n],\qquad {\mathbb{E}}_f[H_n]\leq{\mathbb{E}}_g[H_n].$$

The conjecture formalizes the intuition that stronger reinforcement toward high-degree vertices produces shallower preferential attachment trees. It asserts simultaneous monotonicity for both insertion depth and height under growth-ratio dominance; the supplied source gives no resolution, so the claim remains open.