Converse characterization of motivic weight bounds by singular cohomology

Let $M$ be an object of the rationalized category of effective geometric motives $DM^{\mathrm{eff}}_{\mathrm{gm}}{}_{\mathbb{Q}}$, and let $a,b$ be integers. For every integer $l$, consider the singular cohomology $H^l\otimes\mathbb{Q}(M)$ and its weights.

Converse weight-bound conjecture. If, for all $l$, the weights of $H^l\otimes\mathbb{Q}(M)$ lie between $l+a$ and $l+b$, then

$$M\in DM^{\mathrm{eff}}_{\mathrm{gm}}{}_{[a,b],\mathbb{Q}}.$$

This is the converse to the preceding implication that membership in the weight range $DM^{\mathrm{eff}}_{\mathrm{gm}}{}_{[a,b],\mathbb{Q}}$ forces the cohomological weights to lie between $l+a$ and $l+b$. The supplied text does not state whether this converse has been proved, so its status is recorded as open.