Equivariant smoothing conjecture for antisymplectic cusp singularities

Let $(p\in X)$ be a cusp singularity equipped with an antisymplectic involution $\iota$. Let $D$ denote the dual cycle of the cusp. An equivariant smoothing conjecture. Then $p\in X$ admits an equivariant smoothing if and only if $D$ sits as an anticanonical divisor on a smooth rational surface admitting an antisymplectic involution extending the involution induced on $D$ by $\iota$. This is proposed as an equivariant analogue of Looijenga's smoothability theorem; the necessary and sufficient condition is not proved in general.