Let $V^{2k}$ be a complex subvariety of dimension $2k$ (real dimension $4k$) in $\mathbb C^n$. Let $A$ be a complex $n-k$ dimensional plane in $\mathbb C^n$.

**Question.** Is it true that the inclusion $H_{2n-2k-1}(A\setminus (V\cap A))\to H_{2n-2k-1}(\mathbb C^n\setminus V)$ is injective?

We don't require $V^{2k}$ to be smooth, but $V^{2k}$ must be equidimesional, i.e. all its irreducible components have dimension $2k$.