As recalled above, $K_{X_{t}}$ is torsion for each fixed $t$. Equivalently, $h^{0}(X_{t},rK_{X_{t}})=1$ for some positive integer $r$. Since $t\mapsto h^{0}(X_{t},rK_{X_{t}})$ is upper semicontinuous in the Zariski topology, it follows that $rK_{X_{t}}$ is trivial for a fixed $r$ independent of $t$. Given any snc model $\pi\colon{\mathcal{X}}\to{\mathbb{D}}$, $\pi_{*}{\mathcal{O}}(rK_{{\mathcal{X}}/{\mathbb{D}}})$ is torsion free of rank one, and hence a line bundle. The choice of a trivializing section yields a holomorphic section $\eta$ of $K_{{\mathcal{X}}/{\mathbb{D}}}$, inducing a holomorphic family $\eta_{t}$ of trivializing sections of $rK_{X_{t}}$ for $t\neq 0$. As a consequence, the family of volume forms $\nu_{t}:=|\eta_{t}|^{2/r}$ has analytic singularities at $t=0$, and the result is thus a consequence of Theorem A, since $\mu_{t}=\nu_{t}/\nu_{t}(X_{t})$. ∎