  We prove an equivariant version of the Cosmetic Surgery Conjecture for
strongly invertible knots. Our proof combines a recent result of Hanselman with
the Khovanov multicurve invariants $\widetilde{\operatorname{Kh}}$ and
$\widetilde{\operatorname{BN}}$. We apply the same techniques to reprove a
result of Wang about the Cosmetic Crossing Conjecture and split links. Along
the way, we show that $\widetilde{\operatorname{Kh}}$ and
$\widetilde{\operatorname{BN}}$ detect if a Conway tangle is split.
