Let $C$ be a non-hyperbolic (smooth quasi-projective connected complex algebraic) curve. That is, $C$ is isomorphic to $\mathbb P^1, \mathbb A^1, \mathbb G_m$, or an elliptic curve.
Let $f:X\to C$ be a family of Fano varieties, i.e., $f$ is a smooth projective morphism whose geometric fibres are (smooth projective connected) Fano varieties. (Fano means anti-canonical bundle ample.)
Is $f$ isotrivial? In other words, are all the fibres of $f$ isomorphic?
Note. A family $f:X\to C$ of varieties with semi-ample canonical bundle is isotrivial. This follows from the work of Campana, Kebekus, Kovacs, Lieblich, Viehweg, Zuo, et al.
Note. If $f$ is non-isotrivial, the relative dimension of $f$ will have to be at least three.
Motivation. I think it is reasonable to suspect that certain connected components of the stack of Fano varieties have only finitely many integral points over $\mathbb Z$. If this expectation holds any family of Fano varieties $f:X\to C$, where $C$ is a non-hyperbolic curve, is isotrivial over $C$.