Let $C$ be a smooth connected algebraic curve over $\mathbb C$. Assume that $C$ admits no automorphisms. Let $p_1, p_2 \in C$ be two distinct points.

**Question:** Are the $\mathbb A^1$ homotopy types of $C - p_1$ and $C- p_2$ the same?

I think that the answer is "no", but I do not know how to construct an invariant that distinguishes the two. Of course, the ordinary homotopy types of $C-p_1$ and $C-p_2$ are the same. I also think that the stable $\mathbb A^1$ homotopy types are the same because of the purity cofiber sequence $$C-p_{i} \to C \to (\mathbb A^1, \mathbb A^1 - 0).$$ Is there a way of seeing that these two are not $\mathbb A^1$ homotopic, or of constructing an equivalence between them?