Yet another ideal version of the bounding number

Let $\mathcal{I}$ be an ideal on $\omega$. For $f,g\in {\omega }^{\omega }$ we write $f{\le }_{\mathcal{I}}g$ if $f(n)\le g(n)$ for all $n\in \omega \setminus A$ with some $A\in \mathcal{I}$. Moreover, we denote \mathcal{D}_{\mathcal{I}}=\{f\inω^ω: f^{-1}[\{n\}]\in\mathcal{I} \text{ for everyn\in ω}\} (in particular, ${\mathcal{D}}_{Fin}$ denotes the family of all finite-to-one functions). We examine cardinal numbers $\mathfrak{b}({\ge }_{\mathcal{I}}\cap ({\mathcal{D}}_{\mathcal{I}}\times {\mathcal{D}}_{\mathcal{I}}))$ and $\mathfrak{b}({\ge }_{\mathcal{I}}\cap ({\mathcal{D}}_{Fin}\times {\mathcal{D}}_{Fin}))$ describing the smallest sizes of unbounded from below with respect to the order ${\le }_{\mathcal{I}}$ sets in ${\mathcal{D}}_{Fin}$ and ${\mathcal{D}}_{\mathcal{I}}$, respectively. For a maximal ideal $\mathcal{I}$, these cardinals were investigated by M. Canjar in connection with coinitial and cofinal subsets of the ultrapowers. We show that $\mathfrak{b}({\ge }_{\mathcal{I}}\cap ({\mathcal{D}}_{Fin}\times {\mathcal{D}}_{Fin}))=\mathfrak{b}$ for all ideals $\mathcal{I}$ with the Baire property and that ${\mathrm{\aleph }}_1\le \mathfrak{b}({\ge }_{\mathcal{I}}\cap ({\mathcal{D}}_{\mathcal{I}}\times {\mathcal{D}}_{\mathcal{I}}))\le \mathfrak{b}$ for all coanalytic weak P-ideals (this class contains all ${\mathrm{\Pi }}_4^0$ ideals). What is more, we give examples of Borel (even ${\mathrm{\Sigma }}_2^0$) ideals $\mathcal{I}$ with $\mathfrak{b}({\ge }_{\mathcal{I}}\cap ({\mathcal{D}}_{\mathcal{I}}\times {\mathcal{D}}_{\mathcal{I}}))=\mathfrak{b}$ as well as with $\mathfrak{b}({\ge }_{\mathcal{I}}\cap ({\mathcal{D}}_{\mathcal{I}}\times {\mathcal{D}}_{\mathcal{I}}))={\mathrm{\aleph }}_1$.