Existence of Shelah–Steprāns almost disjoint families in the constructed model

Let ${G}_{{\omega}_{2}}$ be a $({\mathbf{V}}, {\mathbb{P}}_{{\omega}_{2}})$-generic filter, and consider the model ${\mathbf{V}}[{G}_{{\omega}_{2}}]$ constructed in the proof of Theorem. A Shelah–Steprāns almost disjoint family is an almost disjoint family of infinite subsets of $\omega$ with the Shelah–Steprāns property.

Existence conjecture. There are Shelah–Steprāns almost disjoint families in ${\mathbf{V}}[{G}_{{\omega}_{2}}]$, necessarily of size ${\aleph}_{2}$.

The preceding theorem establishes that no Shelah–Steprāns almost disjoint family of size ${\aleph}_{1}$ exists in this model, so the assertion concerns existence at the next possible size. The supplied text does not establish whether such a family exists.