Non-vanishing conjecture for Witten–Reshetikhin–Turaev invariants

Let $M$ be an integral homology sphere and let $\tau _\zeta(M)$ denote its Witten–Reshetikhin–Turaev invariant at a root of unity $\zeta$. Non-vanishing conjecture. For every root of unity $\zeta$, one has

$$\tau _\zeta(M)\ne 0.$$

The source proves non-vanishing for roots whose orders have the form $dp^e$ with $d\in\{1,2,3,4,6\}$, but leaves the assertion for arbitrary roots as a conjecture.