By a partial function from $\omega$ to $\omega$ we understand a function $f:dom(f)\to\omega$ defined on an infinite subset of $\omega$.

A family $\mathfrak F$ of partial functions from $\omega$ to $\omega$ is called domain almost disjoint (briefly, DAD) if the family $(dom(f))_{f\in\mathcal F}$ is almost disjoint in the sense that for any distinct partial functions $f,g\in\mathcal F$ the intersection $dom(f)\cap dom(g)$ is finite.

Let $\mathfrak b_a$ be the smallest cardinality of a DAD family $\mathfrak F$ of partial functions from $\omega$ to $\omega$ such that for every function $g:\omega\to\omega$ there exists a partial function $f\in\mathcal F$ such that $f(n)\not\le g(n)$ for infinitely many numbers $n\in dom(f)$.

It is easy to see that $\mathfrak b\le\mathfrak b_a\le \mathfrak d$.

Question. Is the strict inequality $\mathfrak b<\mathfrak b_a$ consistent?

Remark. The affirmative answer to this question implies a positive answer to this problem. Namely, the strict inequality $\mathfrak b<\mathfrak b_a$ implies the exitence of a monotone cofinal map $\omega^\omega\to\mathfrak b^{\mathfrak b}$.