  It has been recently conjectured by Boyer-Gordon-Watson that a closed,
orientable, irreducible $3$-manifold $M$ is a Heegaard Floer $L$-space if and
only if $\pi_1(M)$ is not left-orderable. In this article, we study this
conjecture from the point of view of lattice cohomology, an invariant
introduced by N\'emethi which is conjecturally isomorphic to the $HF^+$ version
of Heegaard Floer homology. Using the invariant's combinatorial tractability as
a stepping stone, we produce some interesting quite general families of
negative-definite graph manifolds against which to test the Boyer-Gordon-Watson
conjecture. Then, using horizontal foliation arguments and direct manipulation
of the fundamental group, we prove that these families do indeed satisfy the
conjecture.
