By a theorem of Livesay, the 3-sphere has a unique (up to equivariant diffeomorphism) smooth involution with exactly two fixed points. Thinking of $S^3$ as the unit sphere in $\mathbb{R}^4$, this involution is given by $(x,y,z,w) \to (-x,-y,-z,w)$. This same formula gives a smooth involution on the 4-ball with a pointwise fixed arc. Are there any other (orientation reversing) smooth involutions on $D^4$ with a pointwise fixed arc (up to equivariant diffeomorphism)?

I know there are infinitely many distinct involutions with a pointwise fixed $D^2$, but I cannot find any literature on the case where the involution is orientation reversing with a fixed arc.