A strong counterexample to the log canonical Beauville--Bogomolov decomposition

For every $d\ge 4$, we construct a $d$-dimensional, log canonical, $K$-trivial variety with the property that two general fibers of its Albanese morphism are not birational. This provides a strong counterexample to the Beauville–Bogomolov decomposition in the log canonical setting. This construction can also be adapted to construct a smooth quasi-projective variety of logarithmic Kodaira dimension 0 whose quasi-Albanese morphism has maximal variation. On the positive side, we show that the Albanese morphism for log canonical pairs with nef anti-canonical class is a locally stable family of pairs.