Uniform diameter bound for normalized metrics on Lefschetz fibres

Let the fibres $X_y$ of a Lefschetz fibration have complex dimension $n\geq 3$, and let $\widetilde{\omega_t}|_{X_y}$ denote the restricted fibre metric at parameter $t$. The normalized fibre metric is $\frac{1}{t}\widetilde{\omega_t}|_{X_y}$.

Uniform diameter bound. There is a constant $C$ such that, independently of $t$ and the fibre $X_y$,

$$\operatorname{diam}\left(\frac{1}{t}\widetilde{\omega_t}|_{X_y}\right)\leq C.$$

This bound is intended to control the geometry of the normalized fibres in the collapsing Calabi–Yau fibration problem. The supplied text does not indicate whether the assertion has been proved or remains open.