Quantum modular depth conjecture for weakly definite plumbed three-manifolds

Let $G$ be an ADE group of rank $r$. Let $M_3$ be a weakly-negative or weakly-positive plumbed manifold with $n$ junction vertices, meaning that the inverse plumbing matrix restricted to the subspace generated by the junction vertices is negative- or positive-definite. Let $\vec{\underline{b}}$ be any allowed boundary condition.

Quantum modular depth conjecture. The invariant $\widehat Z^G_{\vec{\underline{b}}}(M_3;\tau)$ is related to quantum modular forms of depth up to $r\times n$.

This specializes and refines the general quantum modularity expectation. The statement covers arbitrary ADE gauge groups and weakly-negative or weakly-positive plumbed manifolds, and remains open beyond the cases established in the paper.