Homology conjecture for complements of caustics of E7 and E8 singularities

Let $f$ be a real function singularity of type $E_7$ or $E_8$, and consider the complement of the caustic in the parameter space of a versal deformation of $f$. Homology conjecture. For singularities of class $E_7$, respectively $E_8$, the homology groups of this complement satisfy

$$H_i=0\quad\text{for all }i>1,$$

and

$$H_1\cong{\mathbb Z}^5\quad\text{for }E_7,\qquad H_1\cong{\mathbb Z}^7\quad\text{for }E_8.$$

The preceding theorem establishes the numbers of local complement components and lower bounds for the relevant one-dimensional homology, while this conjecture specifies the full homology groups and remains unresolved in the supplied text.