Finiteness conjecture for motives of bounded height

Let $K$ be a number field and let $c>0$. Fix a type $\Phi=(w,(h^r)_{r\in\mathbb{Z}})$ of motives, with $h^r=\dim \operatorname{gr}_{dR}^r$, and choose $a,b$ such that $h^r=0$ unless $a\leq r<b$. For a $\mathbb{Z}$-motive $M$ over $K$, let $h(M)$ denote its logarithmic height. Finiteness conjecture. There are only finitely many isomorphism classes of motives over $K$ of type $\Phi$ of semi-stable reduction such that $h(M)\leq c$. This conjecture generalizes the corresponding finiteness theorem for abelian varieties underlying Faltings's work on the Tate conjecture; its difficulty is that there is usually no moduli space of $\mathbb{Z}$-motives of type $\Phi$.