The pointed Shafarevich conjecture for moduli spaces

Let $\mathcal{M}$ be the relevant moduli stack of polarized smooth proper varieties with semi-ample canonical bundle. Let $U$ be a variety over $\mathbb{C}$ admitting a quasi-finite morphism

$$U\to\mathcal{M}\otimes_{\mathbb{Q}}\mathbb{C}.$$

Let $C$ be a smooth quasi-projective connected curve over $\mathbb{C}$, with $c\in C(\mathbb{C})$, and let $u\in U(\mathbb{C})$.

Pointed Shafarevich conjecture. The set of morphisms $f:C\to U$ satisfying $f(c)=u$ is finite. This predicts finiteness of families over a curve after fixing one fibre over one marked point. The source states special cases, including proper $U$, one-dimensional moduli, curves, and varieties with a quasi-finite period map, but leaves the general case open.