In the remarks on page 153-154 of Stratified Morse Theory, Goresky and MacPherson make a claim that they say follows from the theorem on that page. It seems to be false and I'm wondering if I'm wondering how to fix it. Here is the claim (all varieties are over $\mathbb{C}$):
Let $X \subset \mathbb{P}^N$ be a subvariety of dimension $n$, $H \subset \mathbb{P}^N$ a linear hyperplane, and $Z \subset X$ a subvariety. Suppose that $X - (Z \cup H)$ is a local complete intersection. Then $\pi_i((X-Z) \cap H) \to \pi_i(X-Z)$ is an isomorphism for $i<n-1$.
Here is a recipe for counter-examples: take $X$ to be a smooth variety (so the l.c.i condition is automatic) and $Z=H$. The claim seems to fail because $H$ was not generic, and indeed if we added the assumption that $H$ is generic then the claim would follow from the "furthermore" part of the theorem.
Does it suffice to add a weaker assumption? In particular, is it enough to add the condition that $H$ is transverse to $Z$?