Yet another ideal version of the bounding number b

We give a sufficient and necessary condition when the ideal pointwise
convergence implies the ideal equal (aka quasi-normal) convergence. The
characterization is expressed in terms of a cardinal coefficient related to the bounding number $\mathfrak{b}$. We examine this coefficient for nice ideals.
I would like also to mention two other ideal versions of the bounding number (one introduced by Farkas an Soukup, and another introduced by Brendle).
Moreover, I would like to present some results about  QN-spaces (introduced by Bukovsky-Reclaw-Repicky) and their ideal counterparts.