The answer seems to be positive according to this paper: Barnabás Farkas, Lajos Soukup: *The zero density ideal, cardinal invariants and related forcing problems*.<sup>1</sup>

> **Theorem 2.3.** If $\mathcal I$ is a rare ideal on $\mathbb N$ then $\mathfrak b = \mathfrak b_{\mathcal I}$ and $\mathfrak d = \mathfrak d_{\mathcal I}$.

Just before this theorem the authors mention mention that the ideal $\mathcal Z_0$ of the sets with zero density is a rare ideal.

A similar result is shown for analytic P-ideals in Corollary 5.5 of *More on cardinal invariants of analytic P-ideals* by the same two authors ([arXiv](https://arxiv.org/abs/1002.2192), [eudml](https://eudml.org/doc/32499)). Again, this class of ideals includes $\mathcal Z_0$.

<sup>1</sup>I wasn't able to find whether the paper was published somewhere, but a preprint is available [here](https://www.renyi.hu/~descript/papers/Fa_So_zero.pdf) ([Wayback Machine](http://web.archive.org/web/20120119124441/https://www.renyi.hu/~descript/papers/Fa_So_zero.pdf)). The same paper was also mentioned in this answer: [Are these two quotients of $\omega^\omega$ isomorphic?](https://mathoverflow.net/q/231077#231088)