Convergent subseries of divergent series

Let $mathscr{X}$ be the set of positive real sequences $x=(x_n)$ such that the series $sum_n x_n$ is divergent. For each $x in mathscr{X}$, let $mathcal{I}_x$ be the collection of all $Asubseteq mathbf{N}$ such that the subseries $sum_{n in A}x_n$ is convergent. Moreover, let $mathscr{A}$ be the set of sequences $x in mathscr{X}$ such that $lim_n x_n=0$ and $mathcal{I}_x eq mathcal{I}_y$ for all sequences $y=(y_n) in mathscr{X}$ with $liminf_n y_{n+1}/y_n>0$. We show that $mathscr{A}$ is comeager and that contains uncountably many sequences $x$ which generate pairwise nonisomorphic ideals $mathcal{I}_x$. This answers, in particular, an open question recently posed by M. Filipczak and G. Horbaczewska.