Measure for which lower Ledrappier (box, Minkowski) dimension and Hausdorff dimension disagree?

I assume that an example of the following is well known to experts, but I've had no luck finding it in the literature:

Given a measure $\mu$ on $\mathbb R^n$, let $\mathcal N(\mu, \epsilon, \delta)$ be the minimum number of sets of diameter at most $\epsilon$ needed to cover mass at least $1-\delta$. (In other words, this is the standard covering number, except that we are allowed to ignore a portion of $\mathbb R^n$ of mass at most $\delta$.)

The lower Ledrappier dimension of a measure $\mu$ on $\mathbb R^n$ is $$\underline d(\mu) := \lim_{\delta \to 0} \liminf_{\epsilon \to 0} \frac{\log \mathcal N(\mu, \epsilon, \delta)}{-\log \epsilon}.$$

The Ledrappier dimension is an appropriately relaxed version of the box dimension for measures, and it was introduced in [Ledrappier 1986].

The Hausdorff dimension of $\mu$ is $$d_H(\mu) := \inf\{\dim_H(S): \mu(S) = 1\},$$ where $\dim_H$ is the usual Hausdorff dimension of a set. In much the same way that box dimension of a set is larger than the Hausdorff dimension, we have $d_H(\mu) \leq \underline{d}(\mu)$ for all measures.

There are lots of examples where the Hausdorff dimension and box (Minkowski) dimension of sets in $\mathbb R^n$ disagree, but according to a sentence in a paper by Pesin and Weiss [Pesin, Weiss 1996] , it is "common" for $d_H(\mu) = \underline{d}(\mu)$ to hold.

Is there a (simple?) example of a measure for which $d_H(\mu) < \underline{d}(\mu)?$

In Pesin's Dimension Theory book, he claims to provide such an example, but I can't find it!