Layered-scattering determination of intrinsic invariants of a submanifold

Let $K$ be a closed smooth $k$-dimensional submanifold of the flat torus $T^n$, with $k\leq n-2$. Choose numbers

$$\epsilon_0>\epsilon_1>\dots>\epsilon_{\lceil k/2\rceil}>0$$

so that the Weyl tubes $\mathsf T(K,\epsilon_j)$ are properly contained in a smooth compact connected domain $M\subset T^n$. Assume that the natural homomorphism $\pi_1(M)\to\pi_1(T^n)$ is trivial and that the billiards $P(K,\epsilon_j)=T^n\setminus\operatorname{int}(\mathsf T(K,\epsilon_j))$ are ergodic. Let $\mathcal L_{\epsilon_j}:\partial_1^+(SM)\to\mathbb R$ be the length, or travel-time, functions of billiard trajectories in $N(K,\epsilon_j)=M\setminus\mathsf T(K,\epsilon_j)$, and let $\mathcal L^{\mathsf{av}}_{0\leq j\leq\lceil k/2\rceil}$ denote their averages. The quantities

$$\mathsf Q_\ell=\int_K H_\ell\;\mu_K,\qquad 0\leq\ell\leq\lceil k/2\rceil,$$

are the intrinsic global invariants of the Riemannian manifold $K$ defined by the cited formulae.

Layered-scattering determination conjecture. The averages $\mathcal L^{\mathsf{av}}_{0\leq j\leq\lceil k/2\rceil}$ determine the intrinsic global invariants $\{\mathsf Q_\ell\}_{0\leq\ell\leq\lceil k/2\rceil}$.

The claim links finitely many averaged travel-time measurements to the coefficients in the Weyl tube-volume polynomial, and hence to intrinsic geometry of $K$. The supplied text presents the statement as a conjecture and gives no resolution.