  We show that the hyper-K\"ahler varieties of OG10-type constructed by
Laza-Sacc\`a-Voisin (LSV) verify the Lefschetz standard conjecture. This is an
application of a more general result, stating that certain Lagrangian
fibrations verify this conjecture. The main technical assumption of this
general result is that the Lagrangian fibration satisfies the hypotheses of
Ng\^o's support theorem. Verifying that the LSV tenfolds do satisfy those
hypotheses is of independent interest. Another point of independent interest of
the paper is the definition and the study of the Lefschetz standard conjecture
in the relative setting, and its relation to the classical absolute case.
