Tubular-neighborhood conjecture for Poincaré duality angles

Let $M^m$ be a closed, smooth, oriented Riemannian manifold and let $N^n$ be a closed submanifold of codimension $m-n \geq 2$. Define the compact Riemannian manifold

$$M_r:= M - \nu_r(N),$$

where $\nu_r(N)$ is the open tubular neighborhood of radius $r$ about $N$, with $r$ restricted to be small enough that $\partial M_r$ is smooth. If $\theta_r^k$ is a Poincaré duality angle of $M_r$ in dimension $k$, then the tubular-neighborhood conjecture asserts

$$\theta_r^k = O(r^{m-n})$$

for $r$ near zero. The preceding examples suggest that removing a tubular neighborhood produces Poincaré duality angles tending to zero at a rate governed by the codimension; the general asymptotic behavior remains open.