P8^1 caustic-complement cohomology conjecture

Let $f_0=x^3+y^3+z^3$, and consider the morsification obtained from the perturbation $x^3-3\varepsilon^2x+y^3-3\varepsilon^2y+z^3$ by splitting the critical points with coordinates $(\varepsilon,\varepsilon)$ and $(-\varepsilon,-\varepsilon)$ into imaginary critical points and each of the remaining two critical points into a pair of real Morse points. This morsification has passport $(0,2,2,0)$. Let $U$ be the corresponding component of the complement of the caustic. $P_8^1$ conjecture. The two-dimensional cohomology group of $U$ with coefficients in $\mathbb Z_2$ is non-trivial; specifically, the degree-one cohomology classes detecting permutations of two critical points of the same Morse index are non-trivial, and their cohomological product is non-trivial as well.

The claim concerns the topology of a component of the complement of the caustic for the $P_8^1$ singularity. The supplied text gives the proposed non-triviality statement but no resolution or additional evidence, so its status remains open.