Bass-Quillen conjecture expects that any vector bundle on $U\times \mathbb{A}^1$, to be extended from $U$. Here $U$ is a regular affine scheme. Being affine is an essential part of the conjecture, you can easily build counter-examples on projective varieties like $\mathbb{P}^1$. For example we have a short exact sequence on $\mathbb{P}^1$ of the form $0\rightarrow \mathcal{O}\rightarrow \mathcal{O}(1)\oplus \mathcal{O}(1)\rightarrow \mathcal{O}(2)\rightarrow 0$. You can deform this extension to the split extension. Obviously $\mathcal{O}(1)\oplus \mathcal{O}(1) \ncong \mathcal{O}\oplus\mathcal{O}(2)$. So the resulting vector bundle on the $\mathbb{P}^1\times \mathbb{A}^1$, cannot be extended. I was wondering whether it is possible to put some more conditions on the projective case to make the conjecture true? (maybe something like forcing the vector bundle on $X\times \mathbb{A}^1$ to have isomorphic restrictions to $X\times \{0\}$ and $X\times \{1\}$.)
Edit: As pointed out by Mohan below, in the example I provided, in the deformation of extensions of $\mathcal{O}$ and $\mathcal{O}(2)$, if you look at $X\times \{1\}$ and $X\times \{-1\}$, you get isomorphic vector bundles, which suggests my suggestion above probably doesn't work.
Edit 2: I decided to add the original problem that I had in my mind that lead to this. Instead of writing a new question I'm adding it here. Assume $X$ is a regular variety. Consider three lines in general position in $\mathbb{A}^2$ denoted by $l_1,l_2$ and $l_3$. Assume we have a vector bundle on $X\times l_1 \cup X\times l_2 \cup X\times l_3\subset X\times\mathbb{A}^2$. Is it possible to extend this vector bundle to $X\times \mathbb{A}^2$? (In the case $X$ is affine it follows from Bass-Quillen theorem)