Existence of an evilly normal kernel for sets with the evil twin property

Let $\mathscr{A}$ be a closed set of games with the evil twin property. A pair $(\mathscr{A},\mathscr{K})$, where $\mathscr{K}\subseteq\mathscr{A}$ is a kernel, is evilly normal if, for all $A,B\in\mathscr{A}$, $A+B\notin\mathscr{K}$ if and only if $A\notin\mathscr{K}$ and $B\notin\mathscr{K}$. The evilly normal kernel conjecture. If $\mathscr{A}$ has the evil twin property, then there exists a kernel $\mathscr{K}$ such that $(\mathscr{A},\mathscr{K})$ is evilly normal. The conjecture generalizes the two properties of generalized flowers used in the preceding proof, asserting that the evil twin property alone guarantees the existence of a kernel with the corresponding normality condition.