Uniformizability conjecture for moduli stacks of polarized varieties

Let $h\in\mathbb{Q}[t]$ be a polynomial, and let $\mathcal{M}_h$ be the stack over $\mathbb{Q}$ of polarized smooth proper geometrically connected varieties with semi-ample canonical bundle. A stack is uniformizable if there is a scheme $M$ and a finite étale morphism $M\to\mathcal{M}_h$.

Uniformizability conjecture. The stack $\mathcal{M}_h$ is uniformizable by a quasi-projective scheme over $\mathbb{Q}$. Uniformizability would allow finiteness results proved for schemes to be applied directly to these moduli stacks; the source notes examples but gives no resolution of the general claim.