Properness and universality of refining forcing on a condition

Determine whether refining forcing is proper below some condition, and whether there exists a Borel $\omega$-refining family $B$ such that every refining real in $B$ over a model containing $B$ is universally refining over that model.

Background

The paper proves that refining forcing is not proper below a particular condition and that the forcing collapses the continuum to $\omega$ below some condition. As with true eventually different forcing, the authors leave open whether a suitable restriction could exhibit properness or a corresponding universality property.