Call a Borel set $A \subseteq [0,1]$ *good* if $$0 < \dim(A) \leq \overline{\dim_\text{M}}(A) < 2 \dim(A),$$ where $\dim(A)$ is the Hausdorff dimension of $A$ and $\overline{\dim_\text{M}}(A)$ is the upper Minkowski dimension of $A$. (Note that the second inequality in the string of three inequalities holds automatically.)

Does a compact set $X \subseteq [0,1]$ of positive Hausdorff dimension necessarily contain a subset which is a

goodset?

A positive answer leads to some more questions: Is there a subset of $X$ with the same Hausdorff dimension as $X$ which is also *good*? To what extent can the $2$ in the definition of *good* be reduced while maintaining a positive answer? Is it possible to find a subset for which the Hausdorff and upper Minkowski dimensions agree?

Otherwise, I'm hoping for a counterexample: a set with positive Hausdorff dimension with the property that all positive-dimension subsets have "large" upper Minkowski dimension. Given such an example, would it further be possible to have an $\epsilon > 0$ such that all positive-dimension subsets have upper Minkowski dimension greater than $\epsilon$?