If $\,\Gamma \subset \mathbb R^n$ is $1$–rectifiable, then its Hausdorff measure is equal to its integral geometric measure. That is, $$\displaystyle\mathcal H^1\left(\Gamma\right) = \int_{G\left({1,\mathbb R}^n\right)} \int_K \operatorname{Card}\left(\left\lbrace {y \in \Pi_K}^{-1}\left(\left\lbrace x\right\rbrace \right)\right\rbrace \right)\, {d\hspace{0.125ex}\mathcal H}^1\left(x\right)\, d \hspace{0.125ex}\Theta_{{1,\mathbb R}^n}\left(K\right),$$ where $\operatorname{Card}(S)$ means the number of points in ${S,\Pi}_K$ denotes orthogonal projection onto ${K,\mathcal H}^1$ denotes the one–dimensional Hausdorff measure, $G\left({1,\mathbb R}^n\right)$ denotes the Grassmanian of unoriented lines through the origin in $\mathbb R^n$, and $\Theta_{{1,\mathbb R}^n}$ is the unique (up to suitable constant) finite Borel measure on $G\left({1,\mathbb R}^n\right)$ which is invariant under the action of the orthogonal group.
I would like to know if the following is true: $$\displaystyle\mathcal H^1\left(\Gamma\right) = \int_{G\left({2,\mathbb R}^n\right)} \int_V \operatorname{Card}\left(\left\lbrace {y \in \Pi_V}^{-1}\left(\left\lbrace x\right\rbrace \right)\right\rbrace \right)\, {d\hspace{0.125ex}\mathcal H}^1\left(x\right)\, d \hspace{0.125ex}\Theta_{{2,\mathbb R}^n}\left(K\right),$$
The more general question where the numbers $1$ and $2$ are replaced by $j$ and $k$ with $j<k<n$ is also of interest.