Family band width conjecture

Let $N\to B$ be a fiber bundle with base $B$ a connected closed manifold, and with all fibers diffeomorphic to a fixed connected closed manifold $Z$. Suppose there is no fiberwise positive-scalar-curvature metric on $N$. Write $n-1\coloneq\dim Z\geqslant2$, with $n-1\neq4$. If the fiber bundle $N\times[-1,1]\to B$ admits a metric $g$ whose fiberwise scalar curvature is bounded below by $\sigma>0$, independently of $b\in B$, then the family band width conjecture.

$$\mathrm{dist}_g\{N\times\{-1\},N\times\{1\}\}\leqslant2\pi\sqrt{\frac{n-1}{n\sigma}}.$$

This is the family version of Gromov's band width conjecture, extending the problem from one manifold to a fiber bundle with no fiberwise positive-scalar-curvature metric. The paper proves the estimate for bundles with infinite family $\widehat{A}$-area; the stated general claim remains open.