Compatibility conjecture for the Betti realization of perverse motives

Let $X$ be a smooth $k$-variety. Let $d4b3(X)$ be the category of perverse motives constructed by Ivorra, and let

$$\mathsf{RL}^{\mathscr N}_X:\mathbf{DA}_{\mathsf{ct}}(X)\rightarrow\mathrm{D}^{\mathrm{b}}(\mathscr N(X))$$

be the triangulated functor constructed by Ivorra. Betti-realization compatibility conjecture. The Betti realization $\mathrm{Bti}_X^*$ is isomorphic to the composition

$$\mathbf{DA}_{\mathsf{ct}}(X)\rightarrow\mathrm{D}^{\mathrm{b}}(\mathscr N(X))\xrightarrow{\operatorname{forgetful}}\mathrm{D}^{\mathrm{b}}(\mathscr P(X))\xrightarrow{\operatorname{real}}\mathrm{D}^{\mathrm{b}}_{\mathrm{c}}(X,\mathbf{Q}).$$

The conjecture would imply, by the same universal-property argument used for the point, that the canonical functor from perverse Nori motives to the corresponding motivic category is an equivalence for every smooth $k$-variety. It is presented as reasonable and reachable with current technology; no resolution is supplied.