I would like to know if there exists a smooth structure on the product R⁴ × [0, 1] of euclidean 4-space R⁴ and the unit interval [0, 1], such that for any s ≠ t in [0, 1] the induced smooth structures on R⁴ × {s} and R⁴ × {t} are non-diffeomorphic.

The idea is to realize at least part of the continuum of distinct smooth structures on R⁴ as a topological continuum.

I would like to know if there exists a smooth structure on the product $\mathbb{R}^4 \times [0, 1]$ of euclidean $4$-space $\mathbb{R}^4$ and the unit interval $[0, 1]$, such that for any $s \neq t$ in $[0, 1]$ the induced smooth structures on $\mathbb{R}^4 \times \{s\}$ and $\mathbb{R}^4 \times \{t\}$ are non-diffeomorphic.

The idea is to realize at least part of the continuum of distinct smooth structures on $\mathbb{R}^4$ as a topological continuum.