Definable E0 classes at arbitrary projective levels

Using a modification of the invariant Jensen forcing of [11], we define a model of ZFC, in which, for a given $n\ge 3$, there exists a lightface ${\Pi }_n^1$-set of reals, which is a ${\mathsf{E}}_0$-equivalence class, hence a countable set, and which does not contain any OD element, while every non-empty countable ${\Sigma }_n^1$-set of reals is constructible, hence contains only OD reals.