Viehweg–Zuo conjecture on non-compact Jacobians of high-genus families

Let $f:X\to\mathbb{P}^1$ be a non-isotrivial family of semi-stable curves of genus $g\geq 1$, and let $s_{nc}$ be the number of singular fibers whose Jacobians are non-compact.

Viehweg–Zuo conjecture. One has

$$s_{nc}\geq 5$$

for $g\gg0$.

A lower bound $s_{nc}\geq4$ is known for all genera, and examples with $s_{nc}=4$ exist only for genera $1$, $2$ and $3$. The conjecture remains open for sufficiently high genera.