Point-cofinite covers in the Laver model

Let $\mathsf{S}_1(\Gamma,\Gamma)$ be the statement: For each sequence of point-cofinite open covers, one can pick one element from each cover and obtain a point-cofinite cover. $\mathfrak{b}$ is the minimal cardinality of a set of reals not satisfying $\mathsf{S}_1(\Gamma,\Gamma)$. We prove the following assertions:

1. If there is an unbounded tower, then there are sets of reals of cardinality $\mathfrak{b}$ satisfying $\mathsf{S}_1(\Gamma,\Gamma)$.
2. It is consistent that all sets of reals satisfying $\mathsf{S}_1(\Gamma,\Gamma)$ have cardinality smaller than $\mathfrak{b}$.

These results can also be formulated as dealing with Arhangel'skiĭ's property $\alpha_2$ for spaces of continuous real-valued functions.

The main technical result is that in Laver's model, each set of reals of cardinality $\mathfrak{b}$ has an unbounded Borel image in the Baire space $\omega^\omega$.