Gukov's conjecture on the depth of quantum modular forms from plumbing graphs

Let $T$ be a plumbing graph (tree) with $r$ nodes of degree at least three. Let $\widehat{Z}_{\boldsymbol{a}}(q)$ denote the associated invariant, and let $\widehat{Z}(q)$ denote the corresponding invariant for a unimodular plumbing matrix.

Gukov's conjecture. The function $\widehat{Z}_{\boldsymbol{a}}(q)$ is a depth $r$ quantum modular form whose quantum set is a subset of $\mathbb Q$. Moreover, for any unimodular plumbing matrix, $\widehat{Z}(q)$ is quantum of depth $r$ with quantum set $\mathbb Q$.

This conjecture generalizes the paper's results for the considered six-vertex $H$-graphs and reformulates Gukov's conjecture on the quantum modularity of $\widehat{Z}_{\boldsymbol{a}}(q)$ and $\widehat{Z}(q)$. The supplied context does not state whether the conjecture has been resolved.