Kuznetsov's reconstruction conjecture for instantons on index-2 Fano threefolds

Let $X$ be a Fano threefold of index $2$, and let $E$ be an instanton on $X$. A line is jumping for $E$ if its restriction to $E$ is nontrivial. The associated jumping-line locus carries a theta-sheaf.

Kuznetsov's reconstruction conjecture. The generic line is not jumping, and $E$ can be reconstructed from its theta-sheaf.

The conjecture concerns the description of instanton bundles through their jumping lines and theta-sheaves. The supplied status evidence says that the first part was reinterpreted and the second part was proved, so the combined statement is solved.