Non-vanishing conjecture for the eighth WRT invariant

Let $M$ be an integral homology sphere, let $\zeta _8$ be a primitive eighth root of unity, and define $\widetilde{\tau }_8(M)=\sqrt{-1}^{\lambda(M)}\tau _8(M)$. Eighth-invariant non-vanishing conjecture. The normalized invariant satisfies

$$\widetilde{\tau }_8(M)-1\in\operatorname{Span}_{\mathbb{Z}}\{4,2\sqrt{2}\}.$$

The source reports computational evidence for this stronger restriction; the preceding proposition establishes only containment in $2(\zeta _8-1)\mathbb{Z}[\zeta _8]$.