Bounding and dominating numbers for preordering $\leq^*$ on bijections $f:\omega\to\omega$

Question

For functions $f, g:\omega\to\omega$ we write $f \leq^* g$ if $\{x\in\omega: f(x)> g(x)\}$ is finite.

Let $S_\omega$ denote the collection of bijections $\varphi:\omega\to\omega$ Similarly to the bounding number and the dominating number respectively, we define

${\frak b}^{\text{bij}} = \min\{|B|: B\subseteq S_\omega \land \forall f\in S_\omega\; \exists b\in B(b\not\leq^* f)\}$, and

${\frak d}^{\text{bij}} = \min\{|D|: D\subseteq S_\omega \land \forall f\in S_\omega\; \exists d\in D(f\leq^* d)\}$.

Do we have ${\frak b}^{\text{bij}}={\frak b}$? And what about ${\frak d}^{\text{bij}}={\frak d}$?

Answer 1

Of course $\mathfrak{b}^\mathrm{bij} \geq 2$ and it is not hard to see that $\{id_\omega,f\}$ is an unbounded family if $f$ is the function that permutes each even number with its successor. So $\mathfrak{b}^\mathrm{bij} =2$.

I suspect $\mathfrak{d}^\mathrm{bij}= \mathfrak{c}$ but I don´t have time to check this.