Generic local o-minimality of the open core without monadic second-order arithmetic

Let ${\mathscr{R}}$ be an expansion of $(\mathbb{R},<,+)$, and let ${\mathscr{R}}^\circ$ denote its open core. The structure $({\mathcal P}(\mathbb{N}),\mathbb{N},\in,s)$ is the two-sorted structure consisting of the power set of $\mathbb{N}$, $\mathbb{N}$, membership, and successor. Generic local o-minimality conjecture. If ${\mathscr{R}}$ does not define an isomorphic copy of $({\mathcal P}(\mathbb{N}),\mathbb{N},\in,s)$, then ${\mathscr{R}}^\circ$ is generically locally o-minimal. The claim is stated as a special case of the paper's main conjectural picture connecting noiselessness, Cantor sets, dense $\omega$-orders, and monadic second-order theory.