Zariski-type connectedness conjecture for totally geodesic submanifolds

Let $M$ be an $m$-dimensional Riemannian manifold of positive sectional curvature. Let $N$ and $H$ be totally geodesic closed submanifolds of dimensions $n$ and $h$, respectively, which intersect transversely. Let

$$j_*: \pi_i\bigl(N-(N\cap H)\bigr)\to \pi_i(M-H)$$

be the homomorphism induced by inclusion. Zariski-type connectedness conjecture. The homomorphism $j_*$ is an isomorphism for $i\leq 2n-m$ and an epimorphism for $i=2n-m+1$. The conjecture is presented as a Riemannian-geometric counterpart of Zariski connectedness results in algebraic geometry; the source does not state a resolution or subsequent status.