Ayoub's conservativity conjecture for Betti realization of geometric motives

Let $\mathbf{DM}_{\mathrm{gm}}(\mathbb{C};\mathbb{Q})$ be the category of geometric motives over $\mathbb{C}$ with rational coefficients, and let $\mathbf{D}(\mathbb{Q})$ be the derived category of rational vector spaces. The Betti realization functor

$$H_B:\mathbf{DM}_{\mathrm{gm}}(\mathbb{C};\mathbb{Q})\to\mathbf{D}(\mathbb{Q})$$

sends a motive $M(X)$ to the singular chain complex of $X^{\mathrm{an}}$. Ayoub's conservativity conjecture. The functor $H_B$ is conservative: for any morphism $f:M\to N$ in $\mathbf{DM}_{\mathrm{gm}}(\mathbb{C};\mathbb{Q})$, if $H_B(f)$ is an isomorphism, then $f$ is an isomorphism.

This conjecture would say that Betti realization detects isomorphisms between geometric motives. In the paper it is used as a general conjectural input for conditional results, and no resolution is supplied.