Reconstruction conjecture from the jumping-line curve and line bundle

Let $Y$ be the Fano threefold under consideration, let $E$ be an instanton on $Y$, and let $D_E$ be its curve of jumping lines. Let $\mathcal L_E$ be the associated line bundle on $D_E$. Reconstruction conjecture. The curve of jumping lines $D_E$ together with the line bundle $\mathcal L_E$ determines the instanton. This asserts that the jumping-line data recover the instanton; the paper does not provide a general proof.