# How many closed measure zero sets are needed to cover the real line, really?

This is a refinement of an earlier question. This question assumes familiarity with combinatorial cardinal characteristics of the continuum. For the reader's convenience, I reproduce below the relevant parts.

Let $\mathcal{E}$ be the $\sigma$-ideal generated by closed measure zero subsets of the real line. It is known that $$\operatorname{cov}(\mathcal{N})\cdot\operatorname{cov}(\mathcal{M})\le \operatorname{cov}(\mathcal{E})\le \min\{\operatorname{cov}(\mathcal{N})\cdot \mathfrak{d},\mathfrak{r}\}$$ and that the first inequality is consistently strict (Bartoszynski-Shelah, Ashutosh).

Is the second inequality consistently strict?

Mathias model (i.e., the countable support iteration of length $\omega_2$ of Mathias poset over a model of CH) satisfies $\mathrm{cov}(\mathcal{E})<\mathfrak{b}$ (recall that both $\mathfrak{d}$ and $\mathfrak{r}$ are above $\mathfrak{b}$). Refeer to Bartoszynski-Judah book Set Theory: On the structure of the real line for the citations below.
Mathias forcing adds a dominating real (Lemma 7.4.4). On the other hand, Mathias forcing satisfies the Laver property (Section 7.4A). It is enough to show that any poset $\mathbb{P}$ with the Laver property forces that the closed measure zero sets from the ground model covers the reals in the extension.
Let $\dot{x}$ be a name for a real in $\mathbb{P}$ and $p\in\mathbb{P}$. Let $\dot{h}$ be a name for a member of $\prod_{n<\omega}2^{2n}$ such that $\dot{h}(n)$ codes $\dot{x}\upharpoonright I_n$ where $\langle I_n\rangle_{n<\omega}$ is the interval partition with $|I_n|=2n$. By the Laver property, there are $q\leq p$ in $\mathbb{P}$ and $S\in\prod_{n<\omega}\mathcal{P}(2^{2n})$ such that $|S(n)|\leq 2^n$ and $q\Vdash\dot{h}(n)\in S(n)$ for all $n<\omega$. Now, $C_S:=\{z\in 2^\omega:\forall_{n<\omega}(z\upharpoonright I_n\in S(n))\}$ is a closed measure zero set (coded in the ground model) and $q\Vdash\dot{x}\in C_S$ (elements for this argument are taken from Section 2.6A).
Other example is a countable support iteration of length $\omega_2$, over a model of CH, alternating between Miller forcing and a proper poset with the Laver property that increases $\mathfrak{r}$ (e.g. Silver forcing, which has the Sacks property).