  We settle the last open case of Kuznetsov's conjecture on the derived
categories of Fano threefolds. Contrary to the original conjecture, we prove
the Kuznetsov components of quartic double solids and Gushel-Mukai threefolds
are never equivalent, as recently shown independently by Zhang. On the other
hand, we prove the modified conjecture asserting their deformation equivalence.
Our proof of nonequivalence combines a categorical Enriques-K3 correspondence
with the Hodge theory of categories. Along the way, we obtain a categorical
description of the periods of Gushel-Mukai varieties, which we use to resolve a
conjecture of Kuznetsov and the second author on the birational categorical
Torelli problem, as well as to give a simple proof of a theorem of Debarre and
Kuznetsov on the fibers of the period map. Our proof of deformation equivalence
relies on results of independent interest about obstructions to enhancing group
actions on categories.
