Full faithfulness conjecture for motivic realization on compact motives

Let $k$ be a field, let $\Lambda$ be a coefficient ring, and let $\Gamma_W\in \mathrm{WCT}(k;\Lambda)$ be a Weil cohomology theory for algebraic $k$-varieties. Assume that the commutative $\Lambda$-algebra $\Gamma_W(k)$ is connective and faithfully flat. Let $\mathbf{SH}_{\acute{\rm e}{\rm t},\,\mathrm{ct}}(k;\Lambda)$ be the thick stable sub-$\infty$-category of $\mathbf{SH}_{\acute{\rm e}{\rm t}}(k;\Lambda)$ generated by $\mathrm{M}(X)(n)$ for $X\in\mathrm{Sm}_k$ and $n\in\mathbb{Z}$.

Full faithfulness conjecture. The motivic realization functor

$$\mathrm{R}^*_{W,\,\mathrm{mot}}:\mathbf{SH}_{\acute{\rm e}{\rm t}}(k;\Lambda)\to \mathrm{coMod}_{\mathcal{H}_{\mathrm{mot}}(\Gamma_W)}$$

associated to $\Gamma_W$ becomes fully faithful when restricted to $\mathbf{SH}_{\acute{\rm e}{\rm t},\,\mathrm{ct}}(k;\Lambda)$.

This is a full faithfulness assertion for motivic realization on the subcategory generated by motives of smooth varieties and Tate twists. The supplied text gives the assertion but no information about whether it has been proved or refuted.