A holomorphic motion with non-subharmonic Hausdorff dimension

We present an example of a holomorphic motion of a Cantor set $\lambda \mapsto A_{\lambda }$ for which neither the Hausdorff dimension $\text{dimH}(A_{\lambda })$ nor the lower Minkowski dimension $\text{dimMl}(A_{\lambda })$ is a subharmonic function of~$\lambda$. This answers negatively a question of Fuhrer–Ransford–Younsi, and distinguishes these dimensions from packing and upper Minkowski dimensions, for which subharmonicity holds. All four dimensions are computed explicitly.