  We generically construct a model in which the ${\Pi^1_3}$-uniformization
property is true, thus lowering the best known consistency strength from the
existence of $M_1^{\#}$ to just $\mathsf{ZFC}$. The forcing construction can be
adapted to work over canonical inner models with Woodin cardinals, which
yields, for the first time, universes where the $\Pi^1_{2n}$-uniformization
property holds, thus producing models which contradict the natural
$\mathsf{PD}$-induced pattern. It can also be used to obtain models for the
$\Pi^1_1$-uniformization property in the generalized Baire space.
