We can define a very simple cardinal characteristic in the following way. Recall the relation $\leq^*$ on $\omega^\omega$ defined by $x\leq^* y$ iff $x(i)\leq y(i)$ for all but finitely many $i$. For $x,y\in\omega^\omega$, say that $x$ and $y$ are *comparable*, denoted by $x\parallel y$, if either $x\leq^* y$ or $y\leq^* x$.

Let $\omega^{*\omega}$ denote the set of sequences of natural numbers that diverge to $\infty$. Define

$\mathfrak{cp}=\min\{|C|:C\subseteq\omega^{*\omega},\ (\forall x\in\omega^{*\omega})(\exists y\in C)\, x\parallel y\}$.

It is not hard to see that $\mathfrak{b}\leq\mathfrak{cp}\leq\mathfrak{d}$. On the other hand, since Cohen forcing can add a real in $\omega^{*\omega}$ incompatible with all ground model reals in $\omega^{*\omega}$, we can prove that $\mathrm{cov}(\mathcal{M})\leq\mathfrak{cp}$.

Now, to the questions:

1) Is $\mathfrak{cp}=\mathfrak{d}$?

2) What happens to $\mathfrak{cp}$ in Miller's model?

Perhaps this cardinal invariant is something trivial, but I haven't been able to figure it out so far.