This question is "take 2" of this older one, following a suggestion of Francois Dorais. Consider the following bornologies $\mathbb{D},\mathbb{E}$ on the set $\mathcal{N}$ of all functions from $\mathbb{N}$ to $\mathbb{N}$:

$\mathbb{D}=\{A: \exists f\in\mathcal{N}\forall g\in A\exists m\in\mathbb{N}\forall n>m(f(n)>g(n))\}$. (

**Dominatable**sets)$\mathbb{E}=\{A: \exists f\in\mathcal{N}\forall g\in A\forall m\in\mathbb{N}\exists n>m(f(n)>g(n))\}$. (

**Escapable**sets)

Clearly we must have $\mathfrak{b}=\mathfrak{d}$ in order for these to yield bornomorphic structures on $\mathcal{N}$, and Francois Dorais showed at the above-linked question that the converse holds as well. However, the "tame maps" version of the question is still open:

Suppose $\mathfrak{b}=\mathfrak{d}$. Must there be a

Borel$i:\mathcal{N}\rightarrow\mathcal{N}$ such that the $i$-image of each set in $\mathbb{D}$ is in $\mathbb{E}$ and the $i$-preimage of each set in $\mathbb{E}$ is in $\mathbb{D}$?

It's not even clear to me (per James Hanson's comment below) what the answer is assuming $\mathsf{V=L}$.