  In this paper we study new invariants $\widehat{Z}_{\boldsymbol{a}}(q)$
attached to plumbed $3$-manifolds that were introduced by Gukov, Pei, Putrov,
and Vafa. These remarkable $q$-series at radial limits conjecturally compute
WRT invariants of the corresponding plumbed $3$-manifold. Here we investigate
the series $\widehat{Z}_{0}(q)$ for unimodular plumbing ${\tt H}$-graphs with
six vertices. We prove that for every positive definite unimodular plumbing
matrix, $\widehat{Z}_{0}(q)$ is a depth two quantum modular form on
$\mathbb{Q}$.
