Looping directions and integrals of eigenfunctions over submanifolds

Let $(M,g)$ be a compact $n$-dimensional Riemannian manifold without boundary and $e_\lambda$ be an $L^2$-normalized eigenfunction of the Laplace-Beltrami operator with respect to the metric $g$, i.e $$-\Delta_g e_\lambda = \lambda^2 e_\lambda \qquad \text{ and } \qquad \| e_\lambda \|_{L^2(M)} = 1.$$ Let $\Sigma$ be a $d$-dimensional submanifold and $d\mu$ a smooth, compactly supported measure on $\Sigma$. It is well-known (e.g. proved by Zelditch in far greater generality) that $$\int_\Sigma e_\lambda \, d\mu = O(\lambda^\frac{n-d-1}{2}).$$ We show this bound improves to $o(\lambda^\frac{n-d-1}{2})$ provided the set of looping directions, $$\mathcal{L}_{\Sigma} = \{ (x,\xi) \in SN^*\Sigma : \Phi_t(x,\xi) \in SN^*\Sigma \text{ for some } t > 0 \}$$ has measure zero as a subset of $SN^*\Sigma$, where here $\Phi_t$ is the geodesic flow on the cosphere bundle $S^*M$ and $SN^*\Sigma$ is the unit conormal bundle over $\Sigma$.