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

Let $(X,\mathrm{\Delta })$ be a projective, log canonical, $K$-trivial pair over the complex numbers. Let $Z$ be a minimal log canonical center of $(X,\mathrm{\Delta })$ and suppose that there exists a torus $\mathbb{T}\subseteq \mathrm{Aut}(X)$ preserving $\mathrm{\Delta }$ and such that $\dim \mathbb{T}={\mathrm{codim}}_{X}Z$. Then we show that two general fibers of the Albanese morphism ${\mathrm{alb}}_X$ 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.