Why upper support bounds do not prove the bridge

For $R\le t+s$, the time-zero balls

intersect. Indeed, along the segment joining the centers one can choose distances $a\le t$ and $R-a\le s$ precisely when $R\le t+s$. The resulting point

lies in $J^{-}(x)\cap J^{-}(y)$.

This explains the reflected geometry. It does not show that $z\in {\mathrm{supp}}_{\mathrm{fun}}{B}_f$. Membership in the upper set $P_F$ is not membership in the actual support. Likewise, a use of $y\in {\mathrm{supp}}_{\mathrm{fun}}{W}_g$ requires more than $y\in G$.