Given a Lipschitz map between Carnot Groups $ f : G_1 \to G_2$, with homogeneous dilations $ \delta^1_s, \delta^2_s$, we have the almost everywhere Pansu derivative $ D_H f(x)(y) = \lim_{s\to 0} \delta^2_{1/s}[f(x)^{-1}f(x\delta^1_s y)] $. With $Q$ as homogeneous dimension of $G_1$ and Carnot-Caratheodory metrics $ d_1,d_2$, the metric Jacobian is defined as $$ J_f(x) = \liminf_{r\to 0}\frac{\mathcal{H}^Q_{d_2}(f(B_{d_1}(x,r)))}{\mathcal{H}^Q_{d_1}(B_{d_1}(x,r))}$$ We have to show that if $D_Hf(x)$ is not injective then $ J_f(x) = 0 $. Now we easily find using properties of metric and dilations that $ d_2(f(xy),f(x)D_Hf(x)(y)) = o(d_1(0,y))$ for identity $ 0 \in G_1$ and hence taking $ y \in B_{d_1}(0,r)$ hence having $ xy \in B_{d_1}(x,r)$ we find that $ f(B_{d_1}(x,r)) $ lies in an $ o(r)$ neighborhood $N$ of $ f(x)D_Hf(x)(B_{d_1}(0,r))$. According to Ch. 6 of the book entitled Introduction to Heisenberg groups... by Capogna, Danielli, using that Hausdorff dimension of $ f(x)D_Hf(x)(B_{d_1}(0,r))$ is $ \leq Q-1$ and the fact that $$ \mathcal{H}^Q_{d_2}(f(A)) \leq [Lip(f)]^Q \mathcal{H}^Q_{d_1}(A)$$ it follows from a $ \boldsymbol{covering\ \ argument}$ that $ \mathcal{H}^Q_{d_2}(f(B_{d_1}(x,r))) \leq \mathcal{H}^Q_{d_2}(N) = o(r^Q) = o(\mathcal{H}^Q_{d_1}(B_{d_1}(x,r)))$. I can not seem to find the covering argument precisely. Any help would be really helpful. Thank you.
P.S. All the products like $xy$ are group operations on $x$ and $y$.