Suppose $X$ is a completely metrizable space with no isolated points. Let $\mathcal{ND}_X$ denote the ideal of nowhere dense subsets of $X$, and let $\mathcal{M}_X$ denote the ideal of meager subsets of $X$.
Each of these collections of sets can be viewed as a poset, ordered by inclusion. Let $\mathrm{cof}(\mathcal{ND}_X)$ and $\mathrm{cof}(\mathcal{M}_X)$ denote the cofinalities of these two posets. That is, $$\mathrm{cof}(\mathcal{ND}_X) = \min \{|\mathcal F| : \, \mathcal F \subseteq \mathcal{ND}_X \text{ and } \forall A \in \mathcal{ND}_X \,\exists B \in \mathcal F \ \text{s.t.} \ B \supseteq A\},$$ $$\mathrm{cof}(\mathcal{M}_X) = \min \{|\mathcal F| : \, \mathcal F \subseteq \mathcal M_X \text{ and } \forall A \in \mathcal M_X \,\exists B \in \mathcal F \ \text{s.t.} \ B \supseteq A\},$$
Question 1: Is it consistent to have $\mathrm{cof}(\mathcal{ND}_X) > \mathrm{cof}(\mathcal{M}_X)$ for some completely metrizable space $X$ with no isolated points?
If $X$ is a Polish space, then it follows from a theorem of Fremlin that $\mathrm{cof}(\mathcal{ND}_X) = \mathrm{cof}(\mathcal{M}_X)$. Note that if $X$ is Polish, then $\mathrm{cof}(\mathcal{M}_X)$ is one of the familiar cardinal characteristics of the continuum, usually denoted $\mathrm{cof}(\mathcal M)$ (with no subscript). The simplest example of a completely metrizable space $X$ for which I do not know whether $\mathrm{cof}(\mathcal{ND}_X) = \mathrm{cof}(\mathcal{M}_X)$ is the space $X = D^\omega$, where $D$ is the discrete space of size $\aleph_1$.
Question 2: Is it consistent to have $\mathrm{cof}(\mathcal{ND}_X) > \mathrm{cof}(\mathcal{M}_X)$ when $X$ is the countable power of the discrete space of size $\aleph_1$?
