Reconstruction of Fano manifolds from the orthogonal to an exceptional vector bundle

Let $X$ and $Y$ be smooth complex Fano manifolds of dimension at least three with very ample anticanonical bundles and let $V_X$ and $V_Y$ be exceptional vector bundles on $X$ and $Y$, respectively. We prove that a Fourier--Mukai equivalence $V_X^\perp\simeq V_Y^\perp$ implies $X\simeq Y$.