Nonexistence of box-counting dimension for typical connected Julia sets

Let $J$ be a connected Julia set, and let its box-counting dimension mean the limit, when it exists, obtained from the asymptotic scaling of covering numbers by sets of diameter tending to zero. A typical connected Julia set is understood in the informal sense used by the source; the unit circle, Basilica, and Rabbit are mentioned as possible exceptions.

Nonexistence conjecture. The box-counting dimension of a typical connected Julia set does not exist.

The claim is based on the numerical and graphical evidence in the paper. The source explicitly notes that the unit circle is an exception and suggests that certain named examples, including the Basilica and Rabbit, may also be exceptions; no resolution is given.