Equivariant smoothing criterion for antisymplectic cusp singularities

Let $(p\in X)$ be a cusp singularity equipped with an antisymplectic involution $\iota$ fixed point free on $X\setminus\{p\}$, and let $D$ be its dual cycle. Let $Y$ be a smooth rational surface containing $D$ as an anticanonical divisor, and let $j$ be an antisymplectic involution on $Y$. Equivariant smoothing conjecture. The singularity $p\in X$ admits an equivariant smoothing with respect to the $\mathbb{Z}/2\mathbb{Z}$-action induced by $\iota$ if and only if $j$ is fixed point free on $Y\setminus D$ and extends the involution induced on $D$ by $\iota$. This is the paper's proposed necessary-and-sufficient criterion, modeled on the proved nonequivariant criterion for cusp smoothability; it remains open.