Moduli conjecture for A^1-contractible smooth varieties

Let $m\geq 3$ and $n\geq 0$. A connected $n$-dimensional scheme $S$ is given, together with a smooth morphism

$$f:X\longrightarrow S$$

of relative dimension $m$, whose fibers are $\mathbb{A}^1$-contractible. For a fixed field $k$, consider the fibers over the $k$-points of $S$.

Moduli conjecture. For every $m\geq 3$ and every $n\geq 0$, there exists such an $S$ and $f$ for which the fibers over the $k$-points of $S$ are all non-isomorphic.

This predicts arbitrary-dimensional moduli of pairwise non-isomorphic smooth $\mathbb{A}^1$-contractible varieties. The paper constructs explicit families in dimensions at least $6$; the conjectural assertion concerns every relative dimension at least $3$.