The Dolgachev-Weisfeiler affine-fibration conjecture

Corollary 7.1: for the fourfold $X=\mathrm{Spec}\,A$ of the cancellation counterexample, the maps $\pi:X\to\mathbb A^1$ (given by $p$) and $\rho=(p,H):\mathbb A^5\to\mathbb A^2$ are smooth surjective morphisms of relative dimension three whose fiber over every point is $\mathbb A^3$ over the residue field, yet neither is Zariski-locally trivial. Local triviality would make $A$ a locally polynomial algebra over $\mathbb C[p]$ and hence, by Bass-Connell-Wright and Quillen-Suslin, a polynomial ring, contradicting Theorem 1.1. So the conjecture fails in relative dimension three over the smooth bases $\mathbb A^1$ and $\mathbb A^2$. Relative dimension two is not addressed.

Corollary 7.1: for the fourfold $X=\mathrm{Spec}\,A$ of the cancellation counterexample, the maps $\pi:X\to\mathbb A^1$ (given by $p$) and $\rho=(p,H):\mathbb A^5\to\mathbb A^2$ are smooth surjective morphisms of relative dimension three whose fiber over every point is $\mathbb A^3$ over the residue field, yet neither is Zariski-locally trivial. Local triviality would make $A$ a locally polynomial algebra over $\mathbb C[p]$ and hence, by Bass-Connell-Wright and Quillen-Suslin, a polynomial ring, contradicting Theorem 1.1. So the conjecture fails in relative dimension three over the smooth bases $\mathbb A^1$ and $\mathbb A^2$. Relative dimension two is not addressed.