  The recent observation [R. V. Gorbachev et al., Science {\bf 346}, 448
(2014)] of nonlocal resistance $R_\mathrm{NL}$ near the Dirac point (DP) of
multiterminal graphene on aligned hexagonal boron nitride (G/hBN) has been
interpreted as the consequence of topological valley Hall currents carried by
the Fermi sea states just beneath the bulk gap $E_g$ induced by the inversion
symmetry breaking. However, the valley Hall conductivity $\sigma^v_{xy}$,
quantized inside $E_g$, is not directly measurable. Conversely, the
Landauer-B\"{u}ttiker formula, as numerically exact approach to observable
nonlocal transport quantities, yields $R_\mathrm{NL} \equiv 0$ for the same
simplistic Hamiltonian of gapped graphene that generates $\sigma^v_{xy} \neq
0$. We combine ab initio with quantum transport calculations to demonstrate
that G/hBN wires with zigzag edges host dispersive edge states near the DP that
are absent in theories based on the simplistic Hamiltonian. Although such edge
states exist also in isolated zigzag graphene wires, aligned hBN is required to
modify their energy-momentum dispersion and generate $R_\mathrm{NL} \neq 0$
near the DP persisting in the presence of edge disorder. Concurrently, the edge
states resolve the long-standing puzzle of why the highly insulating state of
G/hBN is rarely observed. We conclude that the observed $R_\mathrm{NL}$ is
unrelated to Fermi sea topological valley currents conjectured for gapped Dirac
spectra.
