O'Grady's conditional theorem for numerical $(K3)^{[2]}$ fourfolds

Let $X$ be a numerical $(K3)^{[2]}$ and suppose that Items (1) through (6) of Proposition (hodgeprop) hold. O'Grady's conditional theorem. Item (1) of Theorem (mainthm1) holds. The statement is presented as a consequence under the listed hypotheses, rather than as an open conjecture; the source gives no evidence that these hypotheses or the resulting theorem are unresolved.