Shafarevich conjecture for families of polarized Calabi–Yau varieties

Let $C$ be a smooth projective curve, let $S\subset C$ be a finite subset, and fix an integer $q\geq 2$. Let $\mathbb T$ denote the parameterizing scheme for non-isotrivial families of curves of genus $q$ over $C$ with singular locus $S$. Shafarevich conjecture. The following four assertions hold:

1. The elements of the set of non-isotrivial families of curves of genus $q$ over $C$ with singular locus $S$ are parameterized by points of a scheme $\mathbb T$ of finite type over $\mathbb C$ (boundedness).
2. Any deformation of a non-isotrivial family of curves of genus $q$ over $C$ with singular locus $S$ is trivial; equivalently,

$$\dim \mathbb T=0.$$

(rigidity). 3. No non-isotrivial family of curves of genus $q$ exists if

$$2g(C)-2+\#S\leq 0,$$

equivalently,

$$\mathbb T\neq\emptyset\Rightarrow 2g(C)-2+\#S>0$$

(hyperbolicity). 4. For a non-isotrivial family $f:X\rightarrow C$, $\deg f_*\omega_{X/C}^m$ is bounded above in terms only of $g(C)$, $\#S$, $q$, and $m$ (weak boundedness). This conjecture concerns the finiteness, rigidity, hyperbolicity, and quantitative boundedness of families with prescribed base and singular locus; the supplied source does not indicate whether these assertions have been resolved individually.