Topological invariance and existence conjecture for level-k series

Let $K$ be a knot, let $\zeta$ be a root of unity, and let $\gamma$ be a non-degenerate Neumann–Zagier datum arising from a triangulation of the knot complement. Topological invariance conjecture. For every knot $K$, there exists a triangulation with a non-degenerate Neumann–Zagier datum $\gamma$. The series $\phi_{\gamma,\zeta}(\hbar)$ is independent of the choice of triangulation and $\gamma$, up to multiplication by $\zeta^{1/12}$ and $e^{\hbar/(24k)}$; modulo these ambiguities it equals $\phi_{K,\zeta}(\hbar)$, the series on the right-hand side of the Quantum Modularity Conjecture. Experimental checks for many knots and state-integral invariance motivate the claim, but the assertion is not proved in general.