Vanishing homology and intersection homology under constant Euler characteristic

Let $G=(G_1,\ldots,G_{n-1}):\mathbb{C}^n\to\mathbb{C}^{n-1}$ be a polynomial mapping with $n\geq 2$ and $K_0(G)=\emptyset$. Assume that there exists a very good projection with respect to $t^0\in\mathbb{C}^{n-1}$. Suppose that the Euler characteristic of $G^{-1}(t)$ is constant for every $t\in\mathbb{C}^{n-1}$. Let ${\mathcal{V}}_G(\rho)$ be the associated variety for a real positive function $\rho$. Vanishing homology conjecture. There exists a real positive function $\rho:\mathbb{C}^n\to\mathbb{R}$ such that

$$H_2({\mathcal{V}}_G(\rho),\mathbb{R})=0$$

and

$$IH_2^{\overline{t}}({\mathcal{V}}_G(\rho),\mathbb{R})=0.$$

This is the second suggested conjecture in the discussion and concerns realizing the topological consequences of constant Euler characteristic through a suitable choice of the function $\rho$.