On the birational isotriviality of the Albanese morphism of a log Calabi-Yau pair with a torus action

Let $(X,Δ)$ be a projective, log canonical, $K$-trivial pair over the complex numbers. Let $Z$ be a minimal log canonical center of $(X,Δ)$ and suppose that there exists a torus $\mathbb{T}\subseteq\operatorname{Aut}(X)$ preserving $Δ$ and such that $\dim\mathbb{T}=\operatorname{codim}_X Z$. Then we show that two general fibers of the Albanese morphism $\operatorname{alb}_X$ are birationally equivalent. In particular, the pathological example of a projective, log canonical, $K$-trivial variety whose Albanese morphism is not generically birationally isotrivial, recently constructed by Bernasconi, Filipazzi, Patakfalvi and Tsakanikas, can be avoided under the additional hypothesis that there exists a torus of large enough dimension in the automorphism group of the given pair.