  In one-dimensional disordered systems with a chiral symmetry it is well-known
that electrons at energy $E = 0$ avoid localization and simultaneously exhibit
a diverging density of states (DOS). For $N$ coupled chains with
zero-correlation-length disorder, the diverging DOS remains for odd $N$, but a
vanishing DOS is found for even $N$. We use a thin spinless graphene nanotube
with disordered Semenoff mass and disordered Haldane coupling to construct $N =
2$ chiral chain models which at low energy have two linear band crossings at
different momenta $\pm K$ (two valleys) and disorder with an arbitrary
correlation length $\xi$ in units of lattice constant $a$. We find that the
finite momentum $\pm K$ forces the disorder in one valley to depend on the
disorder in the other valley, thus departing from known analytical results
which assume having $N$ independent disorders (whatever their spatial
correlation lengths). Our main numerical results show that for this
inter-dependent mass disorder the DOS is also suppressed in the limit of
strongly coupled valleys (lattice-white noise limit, $\xi/a = 0$) and exhibits
a non-trivial crossover as the valleys decouple ($\xi/a\gtrsim 5$) into the DOS
shapes of the $N = 1$ continuum model with finite correlation length $\xi$. We
also show that changing the intra-unit-cell geometry of the disordered Haldane
coupling can tune the amount of inter-valley scattering yet at lowest energies
it produces the decoupled-valley behavior ($N = 1$) all the way down to lattice
white noise.
