Non-vanishing conjecture for Pontryagin classes of orientable flat manifolds

Let $\widehat{\mathsf{LS}}_n$ be the orientable flat manifold associated with $\mathsf{LS}_n$, and let $p_i(\widehat{\mathsf{LS}}_n)$ denote its $i$-th Pontryagin class. For a cohomology class with integral coefficients, its mod $2$ reduction is its image under reduction of coefficients modulo $2$.

Pontryagin-class non-vanishing conjecture. For the orientable flat manifolds $\widehat{\mathsf{LS}}_n$, the mod $2$ reduction of $p_i(\widehat{\mathsf{LS}}_n)$ is non-zero whenever $n\geq 8i$.

The preceding results establish corresponding non-vanishing statements in several cases, including $p_1(\widehat{\mathsf{LS}}_n)$ for $n\geq 8$ and $p_2(\widehat{\mathsf{LS}}_n)$ for $n\geq 16$. The general assertion for all $i$ remains open in the supplied text.