O'Grady's I conjecture for birational involutions on deformations of Hilbert schemes of K3 surfaces

Let $X$ be an irreducible symplectic manifold deformation equivalent to $(K3)^{[n]}$, let $(\,\ )$ be its Beauville form, and let $h\in H^{1,1}_{\mathbb Z}(X)$ satisfy $(h,h)=2$. For $\alpha_i\in H^{1,1}_{\mathbb Q}(X)$, assume that $(\alpha_i,\cdot)$ is integration over an effective analytic $1$-cycle, and let $\beta_i\in H^{1,1}_{\mathbb Z}(X)$ be Poincare dual to an effective divisor. Define the reflection $R_h\colon H^2(X)\to H^2(X)$ by $R_h(\gamma)=-\gamma+(\gamma,h)h$. O'Grady's I conjecture. There exists a birational involution $\phi\colon X\dashrightarrow X$ such that, for every $\gamma\in H^2(X)$,

$$H^2(\phi)(\gamma)=R_h(\gamma)-\sum_i(\gamma,\alpha_i)\beta_i.$$

This conjecture is presented as a consequence of the L conjecture in the supplied text; no proof or resolution is given there.