Suppose that P is a Borel subset of Baire $\times$ Baire, such that for every pair $x,x'$ of reals in the horizontal copy of Baire,

if: $x,x'$ are $E_0$-equivalent (that is, $x(n)=y(n)$ for all but finite $n$)

then: the vertical cross-sections $P_x$ and $P_{x'}$ coincide.

Does this imply that $P_x=P_{x'}$ for all $x,x'$ in Baire (or at least for all $x,x'$ in a set $X$ large in the sense of measure or category) ?