More on yet another ideal version of the bounding number

This is a continuation of the paper [J. Symb. Log. 87 (2022), 1065–1092]. For an ideal $\mathcal{I}$ on $\omega$ we denote \mathcal{D}_{\mathcal{I}}=\{f\inω^ω: f^{-1}[\{n\}]\in\mathcal{I} \text{ for everyn\in ω}\} and write $f{\le }_{\mathcal{I}}g$ if $\{n\in \omega :f(n)>g(n)\}\in \mathcal{I}$, where $f,g\in {\omega }^{\omega }$. We study the cardinal numbers $\mathfrak{b}({\ge }_{\mathcal{I}}\cap ({\mathcal{D}}_{\mathcal{I}}\times {\mathcal{D}}_{\mathcal{I}}))$ describing the smallest sizes of subsets of ${\mathcal{D}}_{\mathcal{I}}$ that are unbounded from below with respect to ${\le }_{\mathcal{I}}$. In particular, we examine the relationships of $\mathfrak{b}({\ge }_{\mathcal{I}}\cap ({\mathcal{D}}_{\mathcal{I}}\times {\mathcal{D}}_{\mathcal{I}}))$ with the dominating number $\mathfrak{d}$. We show that, consistently, $\mathfrak{b}({\ge }_{\mathcal{I}}\cap ({\mathcal{D}}_{\mathcal{I}}\times {\mathcal{D}}_{\mathcal{I}}))>\mathfrak{d}$ for some ideal $\mathcal{I}$, however $\mathfrak{b}({\ge }_{\mathcal{I}}\cap ({\mathcal{D}}_{\mathcal{I}}\times {\mathcal{D}}_{\mathcal{I}}))\le \mathfrak{d}$ for all analytic ideals $\mathcal{I}$. Moreover, we give example of a Borel ideal with $\mathfrak{b}({\ge }_{\mathcal{I}}\cap ({\mathcal{D}}_{\mathcal{I}}\times {\mathcal{D}}_{\mathcal{I}}))=add(\mathcal{M})$.