Finite-cover conjecture for A^1-fibrations in smooth affine threefold families

Let $f:Y\to T$ be a smooth morphism from a smooth affine threefold $Y$ onto a smooth affine curve $T$, and suppose that every closed fiber $Y_t$ has an $\mathbb{A}^1$-fibration of complete type. Finite-cover conjecture. There exists a finite covering $T'$ of $T$ such that the normalization of $Y\times_T T'$ has an $\mathbb{A}^1$-fibration. The assertion predicts that, after a finite base change, fiberwise $\mathbb{A}^1$-fibrations globalize on the normalized total space; the supplied text gives the construction in a particular situation but does not state a general resolution.