Algebraic realization conjecture for vector bundles on cellular complex varieties

Let $X$ be a smooth cellular $\mathbb{C}$-variety, let $Gr_r$ be the Grassmannian classifying rank-$r$ algebraic bundles, and let $\mathrm{Vect}^{top}(X)$ denote topological vector bundles on $X$. The comparison map

$$[X,Gr_r]_{{\mathbb A}^1}\longrightarrow\mathrm{Vect}^{top}(X)$$

is induced by complex realization. Algebraic realization conjecture. This map is surjective, respectively bijective. Cellular hypotheses make motivic obstruction theory particularly tractable; the supplied text does not specify which of the two alternatives is intended in a given situation or whether the conjecture is resolved.