The complement conjecture for (cf-)ISLE classes

Let ${\mathscr{G}}$ be a class of finite groups, and let ${\mathscr{G}}^c$ denote its complement among the finite groups. A class is (cf-)ISLE if it has the (core-free) ISLE closure property considered in the paper. Complement conjecture. If ${\mathscr{G}}$ is a (cf-)ISLE class, then ${\mathscr{G}}^c$ is not a (cf-)ISLE class. This is stated as an equivalent formulation of the corresponding property conjecture; the supplied text gives no resolution, so its status remains open.