The isotriviality conjecture for families over special bases

Let $f:X\to B$ be an algebraic smooth family of canonically polarised manifolds with generically maximal variation, and let $B$ be a special quasi-projective manifold, meaning that every smooth compactification $(Y\vert D)$ of $B$ of the specified type is special. Isotriviality conjecture. The family $f$ is isotrivial: all fibres of $f$ are isomorphic. The source relates this formulation to Viehweg's conjectural picture and notes that related formulations have been proved in low dimension and in various cases, while no complete resolution is supplied here.