Suspension conjecture for topologically contractible smooth complex varieties

Let $X$ be a topologically contractible smooth complex variety with a chosen base-point $x\in X({\mathbb C})$. Write $\Sigma^n_{{\mathbb P}^1}(X,x)$ for the $n$-fold ${\mathbb P}^1$-suspension of the pointed variety $(X,x)$. 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. This suggests a subtle relationship between topological contractibility and ${\mathbb A}^1$-contractibility in the ${\mathbb A}^1$-homotopy category, but the assertion remains open.