Weight-zero conjecture for Grothendieck classes

Let $X$ be a smooth quasi-projective variety over $\overline{\mathbb Q}$, and let $j$ and $k$ be integers. Let $H^k_{\rm dRB}(X,\mathbb Q(j))_{\rm Gr}$ denote the Grothendieck classes in the de Rham–Betti cohomology, equipped with the induced weight filtration $W_\bullet$. Weight-zero conjecture for Grothendieck classes. Grothendieck classes live in weight zero, namely,

$$W_{-1}H^k_{\rm dRB}(X,\mathbb Q(j))_{\rm Gr}=0,$$

and the natural injection

$$W_0H^k_{\rm dRB}(X,\mathbb Q(j))_{\rm Gr}\hookrightarrow H^k_{\rm dRB}(X,\mathbb Q(j))_{\rm Gr}$$

is an isomorphism. This conjecture proposes that the weight filtration on Grothendieck classes is concentrated in weight zero; the source introduces it as a conjectural relationship between Grothendieck classes and the weight filtration, without giving a resolution.