Question 1: Are there some publications or preprints that provide $\Bbb A^1$-fibrant replacements of certain classes of (smooth) schemes?

Of course, smooth schemes that are $\Bbb A^1$-fibrant are $\Bbb A^1$-fibrant replacements for themselves. By proposition 3.19 of Morel-Voevodsky's '$\Bbb A^1$-homotopy theory of schemes' and the properties of the Nisnevich topology, one sees that a smooth scheme $X$ over a field $k$ is an $\Bbb A^1$-fibrant object if and only if it is $\Bbb A^1$-rigid, i.e. for which the canonical map $$ Hom_{Sm/k}(U,X)\rightarrow Hom_{Sm/k}(U\times \Bbb A^1,X) $$ is a bijection for every smooth scheme $U$ over $k$.

Example 2.1.10 and the argument after lemma 2.1.11 in Asok-Morel's Smooth varieties up to $\Bbb A^1$-homotopy and algebraic $h$-cobordisms provide examples of such schemes:

• $0$-dimensional $k$-schemes,
• abelian $k$-integral schemes,
• smooth complex integral schemes that can be realised as quotients of bounded Hermitian symmetric domains by actions of discrete groups,
• any open subscheme of $\mathbb G_m$,
• closed integral (smooth) subschemes of $\Bbb A^1$-rigid schemes, and
• product of $\Bbb A^1$-rigid schemes.

Question 1': What are other known classes of $\Bbb A^1$-rigid (smooth) schemes? Are there any known restrictions on $\Bbb A^1$-rigid schemes?

For both questions, I would like to have as many examples as there exist.

PS Schemes are taken to be separated and of finite type over the base field.