Suspension conjecture for topologically contractible smooth complex affine varieties

Let $X$ be a topologically contractible smooth complex affine variety, and let $x\in X({\mathbb C})$ be a chosen base-point. Write $\Sigma^n_{{\mathbb P}^1}(X,x)$ for the $n$-fold ${\mathbb P}^1$-suspension of $(X,x)$. Affine suspension conjecture. There exists an integer $n\geq 0$ such that

$$\Sigma^n_{{\mathbb P}^1}(X,x)$$

is ${\mathbb A}^1$-contractible; in fact, $n=2$ should suffice. The conjecture is motivated by the expected triviality of the Voevodsky motive of topologically contractible varieties and by known results for examples, but it remains open.