Non-uniformizable sets with countable cross-sections on a given level of the projective hierarchy

We present a model of set theory, in which, for a given $n\ge 2$, there exists a non-ROD-uniformizable planar lightface ${\Pi }_n^1$ set in $\mathbb R\times\mathbb R$, whose all vertical cross-sections are countable sets (and in fact Vitali classes), while all planar boldface ${\mathbf{\Sigma }}_{\mathbf{n}}^{\mathbf{1}}$ sets with countable cross-sections are ${\mathbf{\Delta }}_{\mathbf{n}+\mathbf{1}}^{\mathbf{1}}$-uniformizable. Thus it is true in this model, that the ROD-uniformization principle for sets with countable cross-sections first fails precisely at a given projective level.