Hrušák's dichotomy conjecture for fragmented ideals

Let $\mathcal{I}$ be a fragmented ideal. The associated Rothberger number $\mathfrak{b}(\mathcal{I})$ is the least size of a family witnessing the corresponding Rothberger property. Hrušák's dichotomy conjecture.

For fragmented ideals $\mathcal{I}$,

$$\mathfrak{b}(\mathcal{I})=\aleph_1\quad\text{if $\mathcal{I}$ is not gradually fragmented,}$$

and

$$\mathfrak{b}(\mathcal{I})\geq\operatorname{add}(\mathcal{N})\quad\text{if $\mathcal{I}$ is gradually fragmented.}$$

The paper establishes the lower bound for gradually fragmented ideals and proves the value $\aleph_1$ for several important classes of fragmented ideals that are not gradually fragmented. The conjectured dichotomy for all fragmented ideals remains open.