Intersection of $(n-1)$-rectifiable set with hyperplane

Question

Suppose $E$ is a $(n-1)$-rectifiable set in $\mathbb{R}^n$, which means $E$ is a set that has Hausdorff dimension $n-1$ and $H^{n-1} (E) < \infty$ and there is a countable family of Lipschitz maps $f_i: \mathbb{R}^{n-1} \rightarrow \mathbb{R}^n$ such that the union of their image covers $H^{n-1} $- almost all $E$. Given a hyper-hyperplane $H_0$ and consider the collection of planes $C = \{H_0+v: v \bot H_0 \}$. Is for $L^{n-2}$- a.e. hyperplanes $H \in C$, $H\cap E$ necessarily a rectifiable curve? I think this intuitionally makes sense to me but I'm not sure if it is a valid statement.