A *$\kappa$-tower* in $\mathbb{N}^\mathbb{N}$ is a sequence
$\langle a_\alpha : \alpha<\kappa\rangle$ in $\mathbb{N}^\mathbb{N}$
that is $\le^*$-increasing with $\alpha$
and has no $\le^*$-upper bound.

Piotr Szewczak and I need a reference for the consistency of the existence of a $\kappa$-tower in $\mathbb{N}^\mathbb{N}$ for some (any) cardinal number $\kappa>\mathfrak{b}$.

It seems that this consistency (and much more) is established in Peter Dordal's paper on towers, but we guess there is a more elementary and earlier reference for this specific result?

The motivation comes from selection principles (products of Menger spaces).