Isotopic boundary genericity conjecture for Riemannian metrics

Let $M$ be an open smooth $n$-manifold with a Riemannian metric $g$, and let $\alpha:N\subset M$ be a smooth compact codimension-one submanifold. An isotopy $A:N\times[0,1]\to M$ is small when it is a sufficiently small smooth isotopy of the embedding $\alpha$. Isotopic boundary genericity conjecture. There is a small smooth isotopy $A$ such that the geodesic flow $v^g$ is boundary generic with respect to the submanifold obtained by restricting $SM\to M$ to $A(N\times\{1\})$.

The conjecture proposes that boundary genericity can be achieved by a small perturbation of the embedded hypersurface rather than by changing the metric. The surrounding text presents it as the density half of a broader genericity claim and gives no resolution.